Kawaguchiko by highway bus
Easiest route. Shinjuku to Kawaguchiko in 2 hours by direct bus, lake walk, ropeway, return by 19:00. Book the bus a day ahead in peak season.

From Tokyo
Tokyo to Mt Fuji is 100 km and very doable in a day if you pick the right route. These four plans all return to Tokyo before midnight.
Easiest route. Shinjuku to Kawaguchiko in 2 hours by direct bus, lake walk, ropeway, return by 19:00. Book the bus a day ahead in peak season.
Romancecar from Shinjuku to Hakone Yumoto, then the loop: pirate ship on Lake Ashi, Owakudani ropeway, back via Odawara. Fuji view from the ropeway and Ashi.
Limited express Fujikaiyu or Mt Fuji View Express to Shimoyoshida, 10 minute climb to the pagoda, lunch in Kawaguchiko, return on the same line.
Tokyo to Mishima in 45 minutes on Hikari, then bus or rental car to Lake Tanuki, Shiraito Falls and Asagiri highlands. Quieter, drier side of Fuji.
Fuji is hidden by cloud more than half the year. Look at the 7 day visibility forecast the evening before and shift your day trip a day if the score is below 40. The mountain is only 100 km away but it might as well be invisible on a humid summer afternoon.