What can I do to get train tickets for fast trains in Japan cheaper?

While these are mostly not an option for people just visiting Japan, there are a few ways to get cheaper shinkansen tickets. This answer probably belongs on expats, rather than travel; but, the first two options are mildly applicable to travelers.

1) Go to a discount shop. Several of these near Ueno, Shinjuku, and other hubs. They buy/sell tickets, but the discount is usually only 5-10%. If you luck out and the shop is stuck with a soon-to-expire ticket that meets your needs, you might be able to save around 20%.

2) Buy the JR "???", or ticket book. This has 6 tickets of the same route in a 3-month window, at a mild discount (again, only 5-10% I think). If you have three people, you can do a round trip with these six tickets. But, if you are doing a very long round-trip, the Japan Rail Pass will likely beat this deal.

You may be able to get by with no Japanese language for 1) and 2); but, for the next two options, you'll definitely need someone who can speak/read at a high level.

3) Yahoo Auctions. A slightly-less-expensive version of the discount shop, with a more limited supply, and longer lag time between purchase and ticket delivery.

4) The JR East online system (Eki Net) sells some shinkansen tickets for as much as 35% off the normal price. These are for trains with more stops at the smaller stations (and, sometimes they wait for the super-fast train to zip by). You need to register an account, and the page is only in Japanese. I think the other JR companies have a similar system, but I am only familiar with this one. To get the 35% ticket, you pretty much need to book it the morning it goes on sale (one month before the departure). Some smaller discounts are available up until 2 weeks before departure.

For a single trip on the Tokaido Shinkansen (between Shin-Osaka and Tokyo, which includes Shin-Yokohama), the cheapest option is almost certainly the Platt Kodama plan. Things to note:

• The savings are not as big as what you mention in the question (such big discounts simply do not exist in Japan for single trips): instead of 13,810 yen, you pay 10,400.
• You have to use one of the designated Kodama trains, which are the slowest category (the trip takes about three hours and a half).
• Platt Kodama tickets cannot be purchased on the day of departure.
• Contrary to what the English version of the website implies, you can purchase it at Shin-Osaka station. Just go to a JR Central (not JR West!) ticket office and say you want a Platt Kodama ticket; they will know what to do.

While not terribly useful for the OP's case of going from Osaka to Yokohama only, in general the one way to get significantly discounted tickets is to purchase the Japan Rail Pass before arriving in Japan.

For a flat price of ¥29,110 for the 1-week pass, roughly equivalent to a single full-priced Tokyo-Osaka round trip, you can travel all you want on all JR trains throughout the country, including all Shinkansen (except the very fastest Nozomi service) and the Narita Express (N'EX). This extends even to local train services like the JR Yamanote line, although subways and private railways are not covered.