Hello Cherry,
Do you want to just visit Mt.Fuji or climb?
Anyway, board bas will be the easiest way to go to near Mt.Fuji from Nagoya.
Here is the link.
http://bus-en.fujikyu.co.jp/highway/detail/id/11/#timetable
Or, you can get some idea from TripAdvisor.
https://www.tripadvisor.co.uk/ShowTopic-g303147-i9930-k6599498-Fuji_from_Nagoya_and_then_a_climb_of_Mt_Fuji_suggestions-Fuji_Shizuoka_Prefecture_Chubu.html
Perhaps, you also can enjoy the beautiful sight of Mt.Fuji through the window of Shinkansen bullet train if the weather is all right.