Thankfully you're avoiding Golden Week (a week of 4-days worth of national holidays, where tourists
and japanese residents are traveling for the festivities).
Hostels are not a problem, and actually are so popular that there's quite a wide range of selection. In Tokyo, try
Oak Hotel, which provides private single rooms with ensuite bathrooms, free in-room internet, and is a 10min walk from Ueno. No breakfast, but the hostel is one block from a McDonald's (and Yoshinoya!). Booking available online (¥4500-6000).
Depending on where else you want to go there are plenty of options. It's true that you can basically show up in most business-type hotels with no problem in getting a room, but beware that some places may only have large rooms left, and will charge accordingly for the "extra" space. And while you're still technically avoiding Golden Week, I would still advise booking ahead.
But you don't need to worry. Worst case scenario, you can always check into a capsule hotel (which always has availability) for a night (¥2000-3000). gaijindavid's suggestion is also another safety-net, though in Tokyo you should look more for manga cafe's for the overnight stay (¥1000-2000).
If you don't already have it, I really recommend the Lonely Planet guidebook. They have some good stuff on more out-of-the-way places.
Be careful with the Lonely Planet guide (unless they
very recently updated it). I found that some of the listings (business hours and locations) was out-of-date, and wasn't able to see a number of things. If you want a second opinion, I suggest you get the
Rough Guide: Japan, and then city-based guides from other publishers for the cities you visit. I used to be a LP loyalist, but I always carry at least two guides with me, and for my time in Japan, the Rough Guide proved to be more consistent/informative. However, the LP city guides (Tokyo, Kyoto, etc) are very good.
Oh, and I always take my time in Tokyo at the end of the trip. But that's because I have no willpower, and wouldn't be able to stop myself loading up with goodies I'd have to carry round the country for the rest of the time...
Seconded. You may also consider bringing just one suitcase of necessities, and then buying a second suitcase in Japan for goodies. Bag shops are
everywhere.