Been there a few times. =) Dont know of any fixed gear shops because last time i went i wasn't into it. Ask people @ Fatlace, they should know.
As for place to go
Shibuya - where all the younger people go. lots of clubs, OK shopping.
Roppongi - THE mecca of the night life in Tokyo. Where all the westerners and stationed military people flog to. Also in Roppongi: Roppongi Hills. Very nice shopping center.
Harajuku - all the crazy cosplay shiet happens, very interesting place to vist.
Omotesando - right next to Harajuku, a ultra high end shopping area similar to Santana Row.
Ginza - another high end shopping area but with more 'department store' style, similar to Union Square in SF.
Odaiba (outside of tokyo) - amusement parks, shopping, toyota car museum. its on a man made island and very beautiful.
Hang out at Yoyogi Park (close to Harajuku). They have fixed gear meets & races often.
My personal favorite is a neat little town, where my brother lives, 10min bike ride from Shibuya called Shimo-Kitazawa. They have the coolest in-house brand urban clothes stores and its a chill place to hang out in.
Be prepared to spend a lot of money and have a lot of fun =)
