Can I do a same-day Mt. Fuji hike from Tokyo?

by Dan
(Massachusetts, USA)

I plan to climb on July 2nd and would like to do a same-day hike from Tokyo. It looks like I will take the first bus in the morning from Shinjuku to Fuji Subaru Line 5th Station, but I can't figure out how to get back to Tokyo on the same day. Unless we race up and down the mountain I don't think that we can make the last bus back to Tokyo from 5th Station. Also, I'm assuming that the bus on July 2nd won't require an advance purchase because it's not a particularly busy climbing day, is that right? Any help with this would be appreciated.

Bonus question: any chance that we'll need microspikes on July 2nd or no?

Thanks!