"Japanese culture" is an extremely broad subject - you need to refine it a lot more. There are many categories, like its food, music, castles, arts, temples/shrines, nature, gardens, night life, museums, shopping, history, and a lot more.
Visiting a hot spring (onsen) at least once would be an excellent choice. Near Tokyo you can go to some at Nikko, Kusatsu, or Hakone to name a few. A Tokyo Wide Pass would be an excellent way to get around and save money.
If you'd like some hands on experience of Japanese culture, like tea ceremony, flower arrangement, calligraphy and others, this group offers some.
Tokyo actually has a number of historic landscape gardens and parks from its 400 year history. You can see some listings here, or here, or here.
It's up to you to do your research.
This is probably very typical but I would suggest to go to hot springs. You can find artificial one in Odaiba (very close to central tokyo) http://daiba.ooedoonsen.jp/en/ or going to Hakone (very common)http://www.hakoneyuryo.jp/english/
To get into the Hot Springs bath you should follow some rules might be affected your idea.