...stay in/on/at Mount Minobu...

On or at would acceptable--one would usually say on, it the subject were resident in a house or cabin or other shelter; one would usually say at if there were a village or town there, especially if the village or town were named Mount Minobu. I can't think of any locution which would use "in" which would not sound awkward, or just plain wrong.

Thanks, Setanta.

Though it would be really awesome if he lived in a Bond villain like cavern system/bunker while writing these books.

During his 5-year stay in Mount Minobu, he wrote three books.
This would be acceptable only in this weirdly specific case.

If Mount Minobu was a village or town or city I would say that 'in' would be fitting just as we say He lives in Chicago, or she lives in Elk Creek [village]

Even if it was just a mountain, it would be possible to say he lived/stayed in Mount Minobu for five years in a cave.