Which trail are you planning to visit?
I write here some information in case you choose Yoshida Trail which is the most popular one.
From Shinjuku station, Express buses run for Mt. Fuji 5th station almost every hour from morning to evening as limited services only in summer season.
Even if you cannot take the direct one, take a bus for Kawaguchiko Station and then transfer to a route bus to get to the Mt. Fuji 5th station.
It is not essential to reserve a seat in advance.
You can purchase the ticket just before getting on a bus at the ticket counter of Shinjuku Expressway Bus Terminal which is located a two-minute-walk of Shinjuku Station New South Gate.
Please also check a website which has some more detailed information such as route maps and time schedule. http://bus-en.fujikyu.co.jp/highway/