Yes. But be careful of course, timewise, if you have a flight.
Your best resource that I've found for this is Wikitravel's article on Fuji which has a Get in section.
I was there in July but just wanted a view, not a climb, and was coming from Kanazawa on the west coast. We took trains down to Nagoya, and across, and then inland to Fujinomiya. It's fairly easy - at each point just ask someone if unsure, or if you have a JR wifi pass (pick it up at any JR station) we used that to get google maps, and Google tells you exactly which train and routes to take. They're fairly frequent too so if you miss one, it's all good.
I did all that, still had an hour in Fujinomiya, and still made it to Tokyo by 6pm, so it's definitely feasible for you.
You can approach on the north side and south side like I did, but the Wiki article explains all the options in detail.