One of my favorite places in Hakone is the Open Air Museum, where the above photo was taken. This outdoor museum displays lots of modern, abstract and interesting artwork in the surrounding grounds with mountains, vallies and the ocean as the backdrops. I highly reccomend checking this place out if you make it to Hakone.
For more information please see the Japan-Guide.com.

On a clear day (which isn’t often) you can see Fuji-san from some places in Tokyo, like Roppongi Hills Tower, or Yokohama. Also while taking the shinkansen from Tokyo to Osaka, after about 30 minutes from leaving Tokyo you will get a nice view of Japan’s most sacred mountain.If you want to get a little closer I recommend stopping by Hakone or the Fuji Five Lakes (Fujigoko). Climbing season is during the summer months of July and August. If you want to avoid the crowds you should go on a weekday around the middle of July, when Japanese schools are still in session and before the summer holidays. The photo below was taken from the 5th stage of Mt. Fuji looking away from the summit.

Interested in climbing Mount Fuji? That is one of the things I have on my list of things to do before leaving Japan. To be honest, I know close to nothing about this (other than the peak season is July and August. If you want to climb Mt. Fuji, Japan-Guide.com has a climbing guide to Mt. Fuji, which seems like a good place to start.