Altogether, I've spent about two months in Japan over the last year and, being a student, I think I can help you out as regards enjoying Tokyo on a budget.

If you've only got a week and you're strapped for cash, there's not too much reason to leave Tokyo. It's a huge and amazing place, and you can very easily spend days and days wandering around, window shopping and enjoying the general ambience. Strolling through Shibuya and Harajuku on a sunny Sunday is about the best way to introduce yourself to the place, and it doesn't cost a penny to do. The east side of Shinjuku is a little seedier (if one of those pushy 16-year-old guys with the impeccable hair and the monstrously expensive suits tries to "recruit" you, take it as a compliment!), but just as interesting. Akihabara has been mentioned a number of times, and if you've got anything in common with the average *chan user then it'll be more fun than any amusement park; Animate and Gamers are great fun to look around if you're an anime fan, and there are more places to buy and play games than you can ever hope to count. Odaiba is great too, but it's more fun with a significant other (if you're there with someone special, make sure you ride the big ferris wheel just after sundown, it's expensive but it's worth it).

Contrary to what people have said upthread, NOT everything is super-expensive. Certain things are, but you can eat very cheaply if you know what you're doing (basic ramen, soba, udon and curry places, along with surprisingly good convenience store food, will keep you well fed for under $10 a day, and a hearty dinner at a good restaurant can be had very easily for under $20). You can get the train from Asakusa to Shinjuku for Y160 each way - if you think that's expensive then I'd like to move to your country right away because it sure as hell isn't expensive by British standards. As far as shopping goes, some things are horribly expensive (new games, DVDs, CDs, "vintage" clothing), but others can be so cheap it just isn't funny (used games, consoles, PC bits, books). Accomodation can be a big financial drain, but if things get really tight and you're feeling hardcore, you can get a private booth in a manga cafe overnight for about $20 (maybe less, I'm not sure) - in a decent place like Manboo, this buys you comfy chair (comfy enough to sleep in) use of your own PC, TV and PS2, and gives you unrestricted access to a shower and a dizzying array of comics and games.

Regarding money, credit cards are accepted almost universally in the big cities, but if you whip out your plastic beyond a 50-mile radius of the centre of Tokyo it'll be met with a blank stare, so carry cash in the country. As for ATMs, bank and convenience store ATMs can be a pain in the ass, but the Post Office has been unfailingly reliable for me. Find out where the nearest Post Office to your hostel is, and load up there every couple of days.

If you get sick of the city and want a good, cheap day-trip, I highly recommend Kamakura. Like Kyoto, Nikko and so forth, it has a wealth of historical sites and architecture which are an absolute joy to behold. If you go there at any time of the year other than Golden Week, the outlying temples are all but deserted, and it's wonderfully peaceful. It's a very rural-feeling place as well, which is quite welcome after a week in Tokyo. You can get there on the train (Yokosuka line) for Y900 each way, which beats the ever-loving shit out of the haemmoragingly expensive Shinkansen fare you'd have to shell out for a similar day-trip to Kyoto (about $250 for the round trip).

Alternatively, get the train up to Mitake (it's on the Ome line, a branch of the Chuo line that runs west out of Tokyo), and get the funicular railway up Mitake-san. Then strap on your walking boots for the magnificent hike from there to the peak of Odake-san. Tons of wildlife and lots of amazing views (even on a normal, hazy day, you can see most of the western side of the Tokyo sprawl, which is the only way to truly appreciate how BIG it is). Bring a couple of convenience store onigiri for lunch and you've got yourself a grand day out for about $25.

Any questions, just ask away. I might post a few more random thoughts later.