Are you interested in government and politics? If you are, you fit into a pretty niche group of tourists that actually take time on trips to visit the civic centers of foreign countries. Personally, I think governments are fascinating, and I try to see the places where countries are run, if only to get a perspective on where the policy and decisions are being made for the entire nation. Fortunately, in Japan, it's fairly easy to see almost every major Japanese national government building in a matter of hours, seeing as they are all centrally located in downtown Tokyo. It makes for a wonderful walking tour, where you start at one of four stations and make a loop around the civic center until you've seen all the major ministries and agencies.
While most Japanese government buildings do not permit tours, you can get access to Japan's Diet (parliament) building, which makes for a neat experience.
The other great thing about this walking tour is that you can also visit the Imperial Palace, Hibiya Park, or the more modern Toranomon Hills Tower at the start or finish. Also, since this is where Japan's highest ranking government officials work, there are more delicious eateries and cafes than you can count (most of them reasonably priced at lunch time).
If you want to do a walking tour of the national civic center, I recommend starting at either Nagatacho (Hanzomon; Yurakucho; Namboku lines), Kasumigaseki (Chiyoda; Marunouchi; Hibiya lines), Toranomon (Ginza line), or Kokkaigijido-mae (Chiyoda; Marunouchi lines) stations. As you depart the stations, there will be maps pointing out all of the different government buildings. The general boundaries for the civic center are Hibiya Park, the Imperial Palace, Toranomon Hills, and Hotel New Otani.