There are so many free or cheap things to do in Japan, from exploring a neighborhood or even joining in during a local festival. Being friendly and respectful will help make friends and get you invited to some fun events!
Festivals - The vast majority of festivals in Japan are free to experience, from Tokyo to small villages check the local calendars to see what is happening!
Museums - There are many small and medium museums in Japan that have no entrance fees. One of the best examples is the Sumo Museum in Ryogoku Tokyo; it is also right next to the Edo Tokyo Museum which has entrance fees of only $6!
Shrines/Temples - Many shrine and temple grounds are free to explore. Some particularly popular places like Kiyomizu-dera in Kyoto have entrance fees ($3) but others like the beautiful Fushimi Inari Shrine known for its thousands of red tori gates are free to explore!
See also Japan for Free (PDF).
Volunteer Guides - Many cities have volunteer groups that offer their services to guide you around the areas they live. These groups offer everything from set walking tours to local introductions to their favorite places. Bonus, just like everything else in Japan no tipping!
Sister City Organizations - A bit more specific but check in your local town or schools for Sister City Organizations, may offer yearly trips and ways to be active in Japan-American relations. It's also a great way to garner friends in Japan!