I love visiting historical parks whenever I travel. People are dressed up in period clothing and will explain how things worked and what people did. You can ask questions to your heart's content and take lots of pictures.
Here are a handful to get you started.
I hope you will take the opportunity to visit museums while you While you are in Japan for a semester or a year. There are so many fabulous museums - from advertising to ramen to ukiyoe - that you should be able to find one whatever your personal taste. As always, you should check the listings and the homepage to make sure that the museum is open on the day you want to visit.
Here is a guide in English that will help to alert you to upcoming art exhibitions. Artscape International provides reviews and notices of art shows. There is also a database of art museums in Japan that you can search. Take a look at the Grutt Pass (ぐるっとパス） for discounted tickets in the Tokyo area.
Literary Museums 文学館 have a dual function. They provide exhibitions of literary artifacts and usually also have a small research library. Sometimes literary museums are for a particular person (Dazai Osamu) and sometimes they are for the prefecture or city. If you are lucky enough to be in Tokyo, you could spend many saturdays visiting literary museums like this one in Setagaya. There are good guide books to bungakukan, but you can also check this wikipedia article in Japanese for more information. There is also an association of Literary Museums that has a membership list with links to their homepages. If you are interested in a particular author, visiting his/her literary musuem gives you a chance to learn more about that person and see photographs or artifacts from his/her life. Most musuems have at least a little English, including English language pamphlets.
Literary Monuments 文学碑 are often fascinating. You can find them for songs, events, and authors.