Will I have difficulty booking seats on a JR train from Tokyo to Kyoto during Obon?

It's difficult to speculate about seat availabilities, and not very useful since it usually does not change anything to whether or not you can get a seat.

That said, I have found that a good method to estimate how busy travel will be on a particular day is to look at flight availabilities: if it's a very busy day, flights will be booked out long in advance or at least tickets will be expensive. In this case, on August 13 one can see that soem flights are sold out, but most importantly tickets from Tokyo to places beyond Osaka such as Hiroshima and Fukuoka are much more expensive than normal, which indicates strong demand for travel to these destinations and probably means you will face congestion. Some Japanese sources also foresee that the peak period for travel from Tokyo will be August 12-13.

However, plane tickets from Tokyo to Osaka are still at the normal price of about 9,000-12,000 yen (which made me mistakenly believe that this would not be a busy period in an earlier version of this answer). This gives you a simple and economical way to avoid the Shinkansen crowds: don't take the Shinkansen in the first place.

To answer your Question(s) briefly...

...will they already be sold out?

That is a hard question to answer definitively, however, almost certainly not for all services as ??? [same-day tickets] are sold. During peak periods, ????? [Reserved Standing Tickets] are available for Hokuriku/Tohoku/Akita Shinkansen services however the Tokaido Shinkansen (Tokyo - Shin Osaka) does not have that option available.

_{? ??? [Same-Day Tickets] Require purchasing at a JR Midori-no-Madoguchi however in peak periods expect to wait over an hour. The Tokyo Station waiting lines during rush seasons can be up to 3 hours.}

_{? ????? [Reserved Standing Tickets] Are Only available on Hokuriku/Tohoku/Akita Shinkansen services only for Car 4. They are only sold when all reserved seats have been booked.}

Should I forget the JR pass and just book a Nozomi train instead?

That depends on your travel.
Is the sum of all your rail travel in Japan
? 1. on mostly or totally JR services?
? 2. equal to or greater than the price of a JR Pass?
If the answer is NO to both of those questions then it will be cheaper to go without the Rail Pass (however if you are planning to catch the Shinkansen return to Kyoto, then it is almost certainly going to be cheaper WITH the pass).

Will that open my options a bit?

The Nozomi Super Express is the fastest Shinkansen offered on the Tokaido Line Stopping at Tokyo - Shinagawa = Shin Yokohama - Nagoya - Kyoto - Shin Osaka (and sometimes continuing on to Hakata). If time is your enemy it may open up an hour or two but otherwise your options shouldn't change.

Finally, can we book the reserved seats from the JR office at Narita?

The JR ?????? [Midori-no-Madoguchi] is open 6?30 ? 21?45 and is able to process reservations for any JR rail services throughout Japan.

In more depth...

The Obon season is usually the busiest time for travel in the Japanese calendar. Shinkansen are regularly filled past capacity (with special Reserved Car Standing Tickets) available for purchase on each respective day) and Free Seating cars normally have an hour if not longer wait to board (as the cars may be unable to accomodate all the people in the lines for each timed service).

It is suggested that you purchase tickets in advance through either a friend or travel agent. There are reservation systems available, however they are in Japanese and require both a Japanese address and Japanese form of payment. Note that Shinkansen tickets are sold from one month before each scheduled service and in the cases of Obon usually sell most seats within the first 3 hours of reservations opening (as discounts are offered if purchased through JR online systems).

If you are going to try your luck with the Free Seating car (good luck), a tactic I use often is to wait in line to Car 4 of the Nozomi (or the adjacent Reserved Seating car to the Free Seat car of any other Shinkansen) and cut inside the carriage to grab a seat. This wont win you friends but it usually gets you a seat.

Note that you can still book Nozomi Trains with the JR Pass, however you will need to pay the Express Fee (Your Distance Fee will be free as it comes out of the JR Pass).

Checking Seat Availablity

Seat availability can be checked at the following website
JR Shinkansen Seat Availablity - Japanese

The website is in Japanese, however using is as follows.

Month???????|??Day
Hour?????? ? |??Minute
Shinkansen Name
Departure Station??|??Arrival Station

For the Shinkansen Name you will want either the first or second option.

• ??????????????????? [Nozomi - Hikari - Mizuho - Sakura - Tsubame]
  
• ??? [Kodama] (Stops all stations between Tokyo and Shin-Osaka)

For the Departure Station you will want the 1st choice.

• ?? [Tokyo]

And for Arrival Station you will want the 16th choice.

• ?? [Kyoto]

Then click the button the the right ?? [Search]

The next page will show a table as follows.

Empty Seat Information??????????|?Normal Seat ?| ?Green Seat?
___________________________________________________________________
Service Name | Departure Time | Arrival Time | NS Seat | S Seat | NS Seat | S Seat ___________________________________________________________________
eg. Nozomi 25 |?? 10:30 ??| ? 13:06?? | ? ? ? | ??? | ? ? ? | ??

NS : Non-Smoking, S: Smoking
A ? denotes seats are plentiful.
A ? denotes there are fewer that 30 seats available.
A × denotes there are no seats available.
An ??denotes that particular car is not available on the service.

The information on this site is updated live with JR servers and is accurate to at least 5 minutes.

Although you didnt state it, you may wish to consider other options such as Highway Bus, LCC / standard Air travel or even ?? [Express Trains] (This may be up to a 6 hour trip between Tokyo and Osaka) should you fail to get the Shinkansen tickets that you want / as a back-up plan.

I travelled from Tokyo to Kyoto during Golden Week last month (also incredibly busy). On the way back I booked 3 days in advance and was able to get reserved seats in one of the peak slots (the 6pm from Kyoto).

There are trains every 10-20 minutes or maybe even more often. You will almost certainly be able to get a seat if you go to Tokyo station a few days before your departure and buy your tickets.

My recommendation is: Don't worry too much about it.

The Tokaido Shinkansen has more than 200 trains running every day on each direction, with one train every 3 to 4 minutes in peak times. You are not going to get stranded and not be able to go to your destination. Even in the Bon festival season.

In particular, all Tokaido Shinkansen services have non-reserved cars (jiyuu-seki ???), in which seating is first-come first-serve. If you're boarding from the terminus station of your service (most commonly Tokyo or Shin-Osaka), if you're willing to wait two or three trains to leave, you are very likely to be able to get a seat.

Even if not, standing in the non-reserved cars is allowed. If you don't mind standing for 2 hours and a half, you can even get on whatever train you want!

If you really need a reserved seat, consider buying that when you arrive in Japan. Any midori-no-madoguchi (ticket window) will sell you reserved tickets, and you will most likely find a couple of seats in a train if you buy 2 or 3 days in advance. If you must catch train on a specific time with reserved seats, try 5 or 6 days in advance. Even in the peak season.

I don't know much about the JR pass, but if you're willing to ride a Hikari or a Kodama service, you will notice they have even more non-reserved cars, and less people riding them (most people prefer Nozomi trains, as a ticket costs the same than other trains and are much faster).

Also notice that train tickets (including Shinkansen) have a fixed price. In most circumstances it won't make a difference if you buy 10 days in advance or 2 minutes before the train leaves.

Trains are in my opinion the most convenient method of travel for your situation. Plane tickets have to be bought in advance, use yield management, and require you to move from and to the respective airports. Bus tickets are the same, and are also extremely uncomfortable.

Like fkraiem said, an estimation of seat availability is rather difficult, if not downright impossible.

But there are some tips, that might let you sleep easier:

Shinkansen & Seats

Chances might be seat reservations are sold out, however I find it very unlikely that all tickets for a specific date are gone. Since you don't have to actually reserve a seat, you may come by with a regular ticket.

So basically, reserved seats are considered an additional service. Reserved seats are only in so-called "green cars", which can be identified by a matching emblem (see further below for image). If there are no reserved seats available get a normal ticket and arrive early! Regular cars are boarded by the order of arrival at marked zones at the platforms. You are there first - you board first. (This is the way to queue Japanese style. No, this is not enforced by anybody, in Japan this system just works! :-)

For fares have a look here (I recommend the "How to use ..." pdf first)

If no tickets can be purchased on the Internet (I never tried), they have to be bought directly at a Shikansen ticket terminal. You can find them at every major station, e.g. Tokyo Main-Station. (Yes JR office at Narita International will work as well).

The Shinkansen counters inside the ticket gates at the station, usually only sell tickets for this particular day. Outside the ticket gates there are several possibilities to reserve tickets for a specific day. You can find the Shinkansen travel agencies by looking for the green Shinkansen seat sign (have a look here)

JR Pass

If your journey will just take you from Tokyo to Kyoto via Shinkansen, it is very unlikely for the JR Pass to make sense. If you plan a round trip using the Shinkansen (like an additional trip back to Toyko) the JR Pass will cut equal with the normal ticket prices.

There are two things to consider:

1. The pass by itself will not solve any seating problems, since you don't magically have a seat reservation just because you have a pass. If free seats are available you may book a reservation free of charge, though.
2. You will not be able to buy a JR Pass once you are in Japan. You can order it online via various websites or specialised travel agencies sell them.

While a lot of answers have mentioned a number of points I must add one:

Even though Obon (alongside the New Year and Golden Week) is one of the three busiest times of the year for travel across Japan, that typically only applies to the first and last day!

In 2017, I had no problem turning up at Kyoto station to go to Yokohama for a day trip and back. Obon season was Friday to Wednesday, I travelled on Monday and had empty trains both ways.

Also in 2017, I needed to get from Tokyo to Kyoto on or the day before New Year’s Eve which marked the beginning of the New Year holiday period (can’t remember if it was the Saturday or the Sunday). Trains were packed, sold out for hours in advance and queues at Tokyo station were quite something. This was at the very beginning of the holiday season.

A friend of mine did the same journey a day later and had absolutely no problem finding a seat. On his return trip on the third (the last day of New Year holiday season), trains were again sold out all the way (we actually checked on the second and there was practically nothing left).

This year during Obon, I again travelled Kyoto–Tokyo return. My outbound trip was Friday evening before the holiday period and the only reserved seat option available was about half an hour after I arrived at the station and the middle seat of a group of three in a Hikari (slower train). I couldn’t even choose, it just said ‘few seats left’. I chose my return trip to be a day before the Obon season ended and had the standard choice of seats.

Note that on the Tokaido Shinkansen you must use Hikari (or Kodama but you don’t want to use a Kodama) if you are using a JR pass. Hikari trains are slower and thus less popular with locals travelling between Osaka, Kyoto, Nagoya, Yokohama and Tokyo meaning they tend to have more free seats available. But while Nozomi trains run every ten minutes with a couple of additions in between, there are only two Hikari per hour.