You can take a bus to Mount Fuji from Shinjuku station. However, if you could tell me what station would you like to stop by?
Most people go to the Subaru Fuji 5th Station when they said they're going to Mt. Fuji. However, the Subaru Fuji 5th Station is only reachable by car or bus.
During the climbing season (July - Early September), there's a bus that will take you straight from JR Shinjuku Station - Subaru Fuji 5th Station. However, during offseason, the bus from Shinjuku Station would take you to Kawaguchiko Station, and then you need to take another from Kawaguchiko Station to Subaru Fuji 5th Station.
The bus from Shinjuku to Kawaguchiko station is operated by Fujikyu Bus and Keio Bus
The easiest website to book is the Highway bus (Keio) https://highway-buses.jp/
One way would cost you around 1,750 yen. You should book this beforehand because you can't buy the ticket on the spot.
The price for buses is mostly similar. However, if you want to find easier options (navigating these could be quite annoying) you could take a bus tour to Mt. Fuji.
For the bus that will take you from Kawaguchiko Station to Subaru Fuji 5th Station would cost you around 1,500 yen, and there's one bus on every hour.
JFYI, you can still see Mt. Fuji from Kawaguchiko (or the Fuji 5 Lakes) if the weather is clear. There are buses that would take you to the Five Lakes from Kawaguchiko Station.
If you want, you could also ride a train from Shinjuku Station to Kawaguchiko station. A one-way trip would cost you around 2,500 yen and would take about 2.5 hours.
Hope this helps! Good luck!