Not sure what previous experience you have with Japan, if any. If baseball and/or soccer are your thing, I strongly recommend catching some NPB and J-League games. There's plenty of teams in and around Tokyo, great atmosphere, and most of their stadiums have easy access from public transit (I write with a hard side-eye toward the Seibu Dome). A nice day out is to go wander around the Meiji Jingu shrine in Shibuya before making the short trip to Meiji Jingu Stadium for a Swallows game. There's also a top-tier sumo tournament taking place in Tokyo from
September 10-24.
There are your established tourist sites -- the temples in Asakusa, Imperial Palace, National Museum, Tokyo Tower, Ueno Park, Tokyo Disney (I recommend Disney Sea), the Ghibli Museum (get tickets
well in advance), etc. -- but it also depends on your interests. Into pop/geek culture? Check out Akihabara and Nakano Broadway. Want to sample the bar culture? Roppongi it is. Want some top-tier people-watching? Stop at Harajuku station on a Sunday morning and marvel at the cosplayers. Just want to soak in the Tokyo-ness of it all? Wander around Shibuya and Shinjuku at night (but maybe steer clear of Kabukicho, just to be safe).
As for food, you can eat very well in Tokyo on not a lot of money. The convenience stores alone have sustained many a broke college student. I am a card-carrying
Famichiki (Family Mart's fried chicken) sellout, but Lawson and Seven-11 also have plenty of good, cheap options. The fast food joints are worth a look, too. The likes of McDonalds and Burger King will have unique offerings, sure, but consider going fully local -- Yoshinoya, Matsuya, Sukiya, MOS Burger, Lotteria, CoCo Ichibanya, etc.
A prepaid Pasmo or Suica card will make getting around on trains easier and more affordable than buying a ticket each time. No sense splashing out on a Japan Rail Pass unless you're taking multiple shinkansen trips. I'd also recommend learning some
survival phrases in Japanese, even if it's just 'please', 'thank you', and 'where is the toilet?'. There should be a decent standard of English, especially in the more touristy spots, and you can get by in Tokyo with no Japanese since most signs also have English, but people will appreciate your making the effort.