[Nyororin]

It sounds like you are looking for serious mountain hiking, but I have to say that for a short trip it seems a bit of a waste to spend it looking at, well, nothing but trees similar to what you would see elsewhere.

My personal suggestion would be to look into some of the older pilgrimage routes. Kumanokodo is probably the most well known, and has some absolutely beautiful stretches. It presents not just nature but a good slice of history and tradition along with it. As it was a route for those on pilgrimage, it is split into manageable chunks for single day hikes. The photos up on Wikipedia present it as a long road, but that`s a very small part. The majority of it is following small trails through the mountains.

If you are experienced and know how to keep from getting lost - Aokigahara is also a beautiful place to hike and one of the few locations you will find in central Japan that is pretty much untouched by man.