you can see Fuji-san from Hakone, if the day is clear
if you intend to go on the hakone round course, my feeling is that you should go clockwise instead of anti clockwise.
the anti clockwise route is more convenient and thus more crowded
for clockwise route you will need to leave the train station and take a bus to moto-hakone, walk to the temple and visit the tori gate, walk back to noto-hakone and have lunch there, then take the pirate ship across lake ashi.
book a ryokan along the route, so that you do not need to rush, the next day you can go to gotemba for shopping. i believe there is a nikon shop there as well.
and then back to tokyo.