The bike culture center is part of the Science Museum in Kitanomaru Park close the Yasukuni Jinja and the Imperial Palace.
Look at the color of the flooring on their homepage and you get the idea how interesting it could be there.
However, the Science Museum is not a bad one for kids. Lots of displays to play with and physic/chemical experiment show time. If you plan to go there for the Science Museum that's fine, but I wouldn't go there for the bike culture center only. It ranks second behind this one:
Actually I first went there when I first came to Japan. As i needed to know about the MTB XC racing scene here and how to transfer my racing license over.
Massive wealth of information and its more a libary than an actual museum pretty much have every guide book, route book and magazine on file for the past 50 odd years! (Slight exageration)
But if you are looking for a bike museum then I would suggest Shuzenji UCI Continental Cycling Centre. They have a mueseum there with some pretty ancient bikes. The entry fee will also allow you to ride on the Pista track and also the Continental curcuit that is used in the UCI Tour of Japan.
Oh, definitely the Parasite Museum.
Apart from the wonderful displays, you can also watch young Japanese couples enjoying their first date there. Of all places.
Otherwise I personally would prefer to visit bigger size bike stores rather than visiting one of the bike museums. The Shuzenji Cycle Sports Center (http://www.csc.or.jp/) has some displays, but it isn't worth the trip from Tokyo unless you plan to do some others things there as well.
You can ride the 5 km circuit, a demanding one, but be aware that some of the downhill sections are cordonded off as you can see in the video (http://www.youtube.com/watch?v=a-3ZPG0FVTQ). The Pista track is open only on Saturday and Sundays.