Is there more you would like to do in Japan than just see Mt. Fuji and be able to tick Japan as a destination off your bucket list?
If there is, then I would recommend taking the time to have a longer break there rather than trying to cram a lot into 2 days and being worried in the build up of the weather was going to derail your plans. Obviously this might mean postponing your Japan plans to a later date.
Otherwise if you are only able to do the 2 day stopover, you could do a one day trip to include Mt. Fuji and Hakone and the other day you could spend exploring say the Shibuya, Shinjuku and Akhibara areas of Tokyo.
Not exactly Typhoon related, but there is something magical imho about Tokyo in the rain - especially at night.
Unfortunately, I am constrained by time and can only do a full day in Japan. I like the idea of doing Fuji with the surrounding areas for most of the day and finding something to do in Tokyo once I come back around 8 pm. Not sure what would be open around that time,