Kamikochi, with its mountainous scenery and large areas of undeveloped nature, is one of the most popular hiking spots in Japan.
Only by bus, either from Matsumoto or Takayama. Not as difficult as it may sound to get here. The Alpico Group runs scheduled buses to between Matsumoto - Kamikochi - Takayama. (the trip from Matsusmoto to Kamikochi takes around 2 hours) http://www.alpico.co.jp/access/express/index_e.html
Hike and enjoy the greenery and mountain air. Enjoy one of the most spectacular mountain views Japan has to offer. There are multiple hikes (mostly flat) available from the Tourist Centre - where the busses drop you off. You can tailor your hikes according to how much time you have to spend there.
Unspectacular choices only.
Along the river are several hiking lodges, most with dining rooms, and a campground renting tent sites. A more expensive option is to pre-book at one of the Ryokan and Onsen in the surrounding area. Bus connections run from the site until about 4 or 5 o'clock.