While Kamikochi itself is unspectacular, this 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 abround 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.
Onward buses to Takayama or Matsumoto; depart 1500h in the afternoon latest.