You can take a bus from Shinjuku Expressway Bus Terminal to Kawaguchiko and then take a local bus operated by Fujikyuko Bus from Kawaguchiko to Mt. Fuji 5th station. The local bus to Mt. Fuji 5th station is actually running a year around. However, there are times when the bus cannot run due to the heavy snow. During the winter, the bus usually does not go to the 5th station. In May, it should be OK, but it is hard to tell you that the bus will run on that day for sure. The bus should be able to take you to the 3rd or 4th station, but it may not be able to go up to the 5th station. Anyway, the best you can do is to make a reservation from Shinjuku to Kawaguchiko and take a local bus to go to Mt. Fuji.
Hope you have anice trip in Japan!
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