If you get the chance, there is nothing that beats putting up a night at a traditional inn or RYOKAN. There are quite a few within Tokyo city, but those that offer better facilities and a more traditional and culturally interesting experience, it is best to try those that are located outside of Tokyo (eg. around Lake Hakone, Nikko or Narita town). The one I stayed at was at Narita (see second pics) and I have done a more extensive write up under my Accommodation Tips in my Narita Pages.
The ryokan is usually family run and the room rate will include breakfast, with an optional dinner and lunch to be included. The food is usually of very high quality and standard at ryokans because the families that run the inns takes pride in their service and hospitality. Part of the unique cultural experience is to sleep on a futon (see pics) on a tatami mat floor.
Another interesting feature is the traditional bath ("furo") which is a delightful experience-similar to an "onsen" (public bath) but on a much smaller scale. The ryokans in Tokyo town are much smaller due to space constraints and therefore may not have furo facilities.
Don't miss this great experience!