Should I pack cold weather clothing JUST for the Mt. Fuji trip, or should I buy some clothing while in Japan?

by Dre
(Alabama, USA)

Hey there,

I was hoping to climb Mt. Fuji in early July, up and down in one day. As I understand it, Japan is rather hot and muggy during this time, so I am packing accordingly (shorts and short sleeves).

My question is, should I pack cold weather clothing JUST for the Mt. Fuji trip, or would you advise that I buy some clothing while in Japan, perhaps someplace near the mountain?

Also, would it be realistic to travel from Tokyo, to Mt. Fuji, climb Mt. Fuji, and back to Tokyo in one day? Are there places to stay near Mt. Fuji that are cheaper than staying in a hut?

I'm a very fresh traveler, so please excuse my naïveté.

Thanks!
Dre