Tokyo is ripe with culture - from ancient temples to cutting-edge technology, you can find something to suit your iterests wherever they lie. Beyond the obvious sushi stop, take some time to enjoy the sense of order, respect, and honor that guide Japanese society.
Food! Sushi, ramen, yakitori, tempura it's all there. Eat your way around the city and stop to see the sights in between. It's tough to do on a tight budget, but the generally inexpensive food can make up for the accommodation expenses.
Tokyo is pretty expansive. Grab yourself a PASMO and take the super modern and clean subway anywhere you need to go. If you're planning to leave the Tokyo area, look into the SUICA which works in a few different Japanese cities and on JapanRail. Fun fact: You can use your SUICA / PASMO like a debit card in 7/11, at certain vending machines, and other unexpected places. It's pretty neat how useful it is!
Literally everything. Sushi is everywhere. So is ramen. My personal favorite is a spot in Asakusa whose name I couldn't tell you (the signs are all in Japanese) but whose zaru soba + tempura don combo is probably one of my favorite lunches on the planet. I *think* it's this place. Also keep an eye out for a sushi-go-round (or kaiten-zushi) which is sushi that rides around the restaurant on little plates on a conveyor belt and you just take whichever ones you want. In the end the wait staff will count up your plates and charge you by the colors and quantity of the plates. It's really fun!
As with most major cities in Asia, most younger locals will speak enough English for you to ask some basic questions, if not more. Hello. Thank you. Goodbye. Please.
Don't limit yourself to Tokyo! Japan is full of amazing cities and towns, each with their own unique culture and attractions. For a good excuse to take the Shinkansen high-speed train, take a trip to Kyoto. You can also explore Osaka, Yokohama, and Hakone (for a view of Mt. Fuji!) in day trips.