The Japanese Alps are split into three main ranges, Minami (southern) Chuo (central) and Kita (northern) Arapusu (alps). They can easily be reached by commuter train from Tokyo (or anywhere else for that matter).
The Kita Alps are gorgous and worth a visit, even if its just to hike, but the climbing is decent, especially in late August. Its been years since I was there (lived there from 1995 to -97 when I was on staff at Keio and the Kanagawa Academy of Science and Technology) but you can take a train to Matsumoto and then a bus to the trainheads. You can also climb in Ogawayama which is tucked into the mountains just south of Tokyo just north of the Chuo and Minami Alps.
The area around Ichinokarasawa is also beautiful but I've only been there in the winter and can't comment on the Summer.
As for cultural, don't miss Kamakura just south of Yokahama. It has a lot of temples in town and tucked into the foothills just around the town. Its fairly beautiful.
If you need info, I have japanese guidebooks and maps for most of the areas.