Is it a problem to book a hotel for X days but not be there for part of the booking?

Assuming you intend and expect to pay for all 7 nights, there is no requirement you actually stay there.

Sure, you can tell the desk you may be away. I would be stunned if they had an issue.

You could also set the "No Service/Privacy" indicator. This is not an uncommon occurrence.

While my answer doesn't wholly apply to the asker's particular case since the nights absent are not at the beginning of the stay, in the more general case for other folks who have a similar question, it is worth noting that if one won't be present to check in at the start of the booking, one definitely should contact the hotel so they don't assume it is a case of no-show and give the reserved room to someone else.

Now, you almost certainly can only expect to keep a room available for your arrival that the hotel has been paid for or where they can ensure payment. No hotel I know of will maintain a reservation past the check-in time without receiving money or holding a credit card number for security, and if you show up for the first time on day two and all the rooms are gone, tough luck. Even a credit card almost certainly can't keep the room available to the second night without some form of check-in.

If you've paid in advance, being a no-show is acceptable and as long as you have notified them you can probably expect to keep the room. Don't do it too early as this puts a burden on them to remember for an extended period. Call on the day of the expected check-in.

If your payment is not in advance and you expect to pay at the end of your stay, there are a few additional considerations. First, you'll almost certainly be expected to provide a credit card for security, often at booking time and certainly no later than check-in time. Second, if you don't check in on the expected day by the expected time, everything I said above about check-in comes into play and you should no longer expect a room to be reserved for you, plus you may additionally be charged one night's stay even if you never showed.

Finally, if you do plan to be away from the room for an extended period (more than a day or two, though you theoretically could be risking problems even then), do inform the hotel. Even with a Do Not Disturb sign placed on your door, they may check rooms periodically, and if a room is completely empty with no belongings they may reasonably assume you have left without notice and check you out, so that when you return you find you have no room. Even belongings may be collected and the room retaken if it goes on for too long.

None of this should be surprising. Hotels make money by getting paid for rooms. Anything that could jeopardize their ability to be paid for any room is something they will take action to prevent. Failure to check in, abandonment of a room where they may not be able to charge the full amount of the stay to a held credit card, fear of a room being damaged so that it requires repair or can't be occupied for a period, and so in.

My answer is not specific to Japan so I'd be interested in any comments or improvement that can be made to this answer, but I think it extremely likely that no matter where you are, not showing up at the expected time to check in could put your reserved room at risk.