I used to lead climbing tours up Fuji for a summer (for military tours)...
We would take a bus up to the 5th station (as far up as a bus can go),
hike up about 1/3 of the way just before dark, stay in a go-ya (mountain hut) and get up @ 4am to finish the climb in time for sunrise....so the huts are an option and another way to experience a slice of Japan.
I agree with an earlier poster...Fuji is not really a climbing mountain (boring!)
but a viewing mountain....having said that nobody mentioned the descent in the scree ("lava gravel") option....you can come down REALLY fast..but be careful and keep you leg flexible...there can be boulders hidden in the scree that can make it easy to turn an ankle or worse if you hit them just right

Edited to add...if you take the scree slope your shoes/boots will fill up with the scree unless you wear gaitors or similar...and the abrasion is somewhat hard on your shoes...