Can I climb Fuji-san Friday Sept 25?

My concern is getting there via Gotemba. I want to start at Yoshisdaguchi at around 22:00hrs. Is this a possiblitiy.

Only if you are well equipped and prepared and can organize a rental car or taxi.

Late September is rather late for climbing Fuji-san. don't forget that it is a real mountain. I think you can find all answers to your questions here: http://www.japan-guide.com/e/e6901.html

I've got to Mt Fuji (Subashiri 5th Station) from Gotemba by car.
It is supposed to be one of the smallest walking trails from 5th station to Fuji. Nevertheless, expect at least a six-hours hard climbing. I've climbed Fuji in the 14th of August (great weather...): 6 hours hours climbing and more 3 1/2 hours to descend (in a rush, in fact).
Eventhough it was hard, I can assure you it was an amazing experience. I recommend it!!!
be aware: beverages (including water) are expensive at the Stations (300-500 yen per bottle, depending on the Station).
Good luck and amuse yourself!!!

Please be aware though that if youre climbing it on Sep 25th, none of the huts may be open. You cannot guarantee getting any supplies or shelter on the way up or down.