'Getting to Mount Fuji' really means getting to the lake town of Kawaguchiko at its base. The single easiest way is the direct highway bus from Shinjuku — and the golden rule is to book it the day before, on a clear-forecast morning, because seats sell out and Fuji is a no-show on cloudy days.
Check the forecast and webcams before you commit — a cloudy day means no mountain, full stop. Reserve the bus online the evening before; same-day seats are scarce in peak season. Arrive mid-morning while the air is clearest, and plan to leave on an early-evening bus to beat the return rush.
The direct highway bus from Shinjuku to Kawaguchiko (~2 hrs, ~¥2,000, reserve ahead) is simplest. Trains via Otsuki and the Fuji Excursion limited express also work.
About 2 hours to Kawaguchiko by direct bus; roughly 2.5 hours by train via Otsuki.
Yes — reserve online, ideally the day before, especially in peak season when seats sell out and you want a clear-weather day.