In May 2009 I attended the Sevens International Conference on Pervasive Computing in Nara, Japan. We made a one-day excursion to Kyoto and spent one week after the conference in Tokyo, including a trip to Hakóne in the Mount Fuji area.
Nara is a small city, located one hour by train from Kyoto and Osaka. Still there are some remarkable cultural sites, including the Todaji temple, home of the largest Buddha statue in the world.
It's hard to find another place in Japan with so many temples and cultural sites. We were guided through a temple by a group of students who tried to improve their English.
A week is hardly enough to see the largest metropolitan area in the world. Be prepared for tiny hotel rooms ;-)
A 2 hour train ride from Tokyo Shinjuku station is Hakone, a town located in the Mount Fuji area. Pretty much everything can be done by foot. There is a ropeway and a boat ride. Don't miss to visit one of the Onsen baths!