Oh Tokyo, you have my heart. Despite the fact that a lot of "normal" things I do in the US are frowned upon culturally in Japan - such as yelling and being a loud jerk, dancing, interrupting men while they are speaking, lol. Seriously, though. They are fine with heavy drinking.
I was there for work, but managed to sneak away to do a few fun things, here's the list:
Tsutaya bookstore, Shibuya (although they are in every neighborhood). Filled with beautiful books, mostly in Japanese but some not. Comics, magazines, stationary, and other fun browsy stuff that you think will be easy to carry home in your suitcase until you end up with 10 books.
Roppongi Hills Club. The Mori family owns a shit ton of real estate in Tokyo - and this club is in a Mori tower. One floor of this tower is the Mori Art Museum, which I will mention next. The Roppongi Hills Club is on the 51st floor, totally gorgeous, stylish, and with amazing 360 views of Tokyo. Day or night it's a must see, and the drinks are nice.
Mori Art Museum. There are a million museums in Tokyo, but I think this is the best. It's contemporary art and the museum itself it quite beautiful. I got to see the Takashi Murakami exhibit, which blew my mind.
Kurosawa Soba Noodles. Named after the Director, the restaurant, food, tableware, everything, is beautiful. Lovely place for a light lunch.
Fuku Yakitori Shop. I've done the research and I'm thinking this is probably one of the best places in Tokyo for Yakitori. Some local friends took me here and I couldn't stop eating. We ate almost everything on the menu. It's a small spot, super cool with a small bar and the BBQ grill all fired up and smoking right in front of you. Sake and Yakitori man, that's some good shit.
Narisawa, a two Michelin star "French" restaurant, although more fusion of Japanese and contemporary American I would say. This dining experience is an EVENT - it's a 3 hour lunch with several courses including bread that bakes in front of you in a stone pot, matcha coated butter that looks exactly like fresh moss, essence of the forest "starters", and wagyu beef that will make you giddy. Read this guy's much more informative post on Narisawa.
Isetan Shinjuku is where the super stylish Japanese shoppers go. Nine floors of spectacularly designed displays, obscure Japanese brands, beauty products, luxe homegoods, you name it. Or you don't name it, because you can't, because it's so unique and f*cking gorgeous you will want to max every credit line you have, even your retirement savings.
Tokyo Station Ramen Alley. So much food. But they have an alley way that is strictly ramen shops. All good ramen shops have little ordering computers, where you make your selection, pay, then a little ticket comes out. Once you are ready to be seated (go to the shops with the longest lines, that's always my plan) hand the ticket to the host, sit, and wait patiently.