Here's the thing first-timers need to know: you don't 'do' Mount Fuji as a day trip by climbing it — the climb is a July–September overnight. What you do is go to the lake region at its foot, mainly Kawaguchiko, for the views, the pagoda and a lakeside soak. Here's what's worth your time.
Fuji is shy. Clear mornings in late autumn and winter are your best odds; summer haze and afternoon cloud usually hide the summit. Check the forecast and go early — the reflection on the lake is calmest and clearest soon after sunrise.
For a day trip from Tokyo: the Kawaguchiko lake views, Chureito Pagoda, lakeside parks and onsen, and Fuji-Q Highland. The actual climb is a July–September overnight, not a day activity.
No — climbing Fuji is an overnight effort and only possible July–September. Day-trippers go for the views from the lake towns instead.