After living there for a while, I always enjoyed leaving the city.
Hiroshima is good for all things Okonomiyaki, ABomb stuff and Baseball. Even if you don't like Baseball, go to a game and sit in the right field bleachers. You'll lose your voice by the end of the game. Miyajima shrine is also a nice getaway on a hot day.
Kyoto is aiight for traditional stuff. Temples, shrines, castles and shit like that.
Beppu hot springs is pretty relaxing if you don't mind getting naked with a bunch of tourists.
Fukuoka along the river at night has some pretty decent ramen and oden stands.
If you really must stay in Tokyo, here are my recommendations:
Kichijoji: really nice park and next to the Ghibli museum (a must see if you like movies like Princess Mononoke or Tonari No Totoro). If you go to the museum, you need to buy tickets ahead of time from any old convenience store. Kichijoji also has a pretty cool shopping district.
Ikebukuro: Not my favorite place, but they have a pretty hilarious mall (Sunshine City) that houses something called "Ice Cream City". Here you'll find pretty much every flavor of ice cream you could imagine. Including Squid Flavor.
Harajuku/Omotesando: Go here on a weekend and people watch.
Shinjuku: Golden Gai. This was a brothel neighborhood before/during and shortly after the war. Some cool old buildings are here amidst the relatively new backdrop of Shinjuku. Best seen at night.
Last bit of advice: Get the F out of Roppongi and stay at a Ryokan. These places are way more comfortable and the food is usually included in the price of a room. The food is usually the best part of it. Expect to pay at least $200/night/person.