I just finished an English-teaching job in Japan (in Fukuoka). Here are a couple suggestions:
Travel: Wherever you are in Japan, there are likely many interesting places in and around your town. If you're into history, learn about it by visiting temples, shrines and museums. If shopping or pop-culture is more your thing, check out your closest Big City for shopping, concerts, clubs, etc. (if you're near Fukuoka, Tenjin is great!)
Eat: This was my main cultural activity. There are so many different kinds of Japanese food, and literally every town has it's own specialty. I couldn't stand the smell of tonkotsu ramen when I got to Japan, but now it's one of my all-time favourite foods!
Get a hobby: There are lots of Japanese hobbies that you can pick up: taiko drumming, various martial arts, flower arranging, etc. If you're teaching at a public school, you might even be able to get involved with one of your school clubs.
Make a friend: There are tons of Japanese people who are studying English and would love the chance to sit down and chat with a native English speaker. Find one and go for coffee and get suggestions from a local.
Good luck! about 9 years ago