Odawara is the nearest Shinkansen station, but it's still around 2 hours minimum to Lake Ashi from there. You can do a rushed tour of the Hakone area in a day, but it's better to stay overnight (search for Gora Kadan for everybody's favorite swanky ryokan).

Orientation: http://www.hakone.or.jp/info/map_en.html

But, as said, don't set your expectations too high because the odds of seeing Mt. Fuji just aren't that good.