The bus is not running to Mt. Fuji 5th station during the winter.
If you would like to see Mt. Fuji, I suggest that you go to Kawaguchiko instead of Mt. Fuji 5th station.
You can see Mt. Fuji beautifully from Kawaguchiko, so a lot of people visit there.
If you are going to Kawaguchiko, you can take a highway bus from Shinjuku Expressway Bus Terminal.
It is 2000 yen for one way and 4000 yen for round trip.
It takes about 1 hour 45 minutes from Shinjuku to Kawaguchiko.
You can book your ticket from the following website.
Have a nice trip to Japan!