Mt. Fuji

I would like to see Mt. Fuji, Im going to be staying in Shibuya most of my time in Japan. What rail way would i take to Mt. Fuji? Will i need to stay the night considering travel time? Any hotel recommendations? Thanks

Hakone:-
http://www.japan-guide.com/e/e5206.html
http://www.japan-guide.com/e/e5206.html
http://www.japan-guide.com/e/e2358_008.html
Kawaguchiko:-
http://www.japan-guide.com/e/e6906.html
http://www.japan-guide.com/e/e6905.html

You can do it as a day trip if you wish.

http://www.japan-guide.com/e/e2172.html

Most people climb mt fuji overnight so they can see the sun rise. So yeah, its fine to do it in one day from shibuya, its only around 2 hours away, just take the chuo line to Otsuki then the Fujikyu line down to Fuji.
Or likely even better just a direct bus from tokyo somewhere.