Hello M v,
Most people who climb Mt. Fuji have never climbed a mountain before so you'll have LOTS of company! :)
Mt. Fuji only "officially" closes, but people climb the mountain all year. I suspect that the closing date is more for protection for the many many people who've never had any outdoor experience at all.
September is the beginning of typhoon season. You definitely do not want to be up there during a typhoon. Just check the weather and make sure no typhoons are around and you should be fine right until the end of September. After that it gets really cold at the top and only experienced mountaineers should attempt the mountain.
Mt. Fuji is dangerous in bad weather because it is completely exposed on all sides. It takes the brunt of all the weather from the ocean to the south. It is so big that it creates its own weather patterns. And since there is no shelter on its flanks the winds get very strong. So you have to watch the weather carefully.
The thing you have to watch, too, at the time of year you want to go, is bus access. Often, with the end of climbing season, the bus service is shut down, too, and getting there without a car is very difficult. If you rent a car you will have no problem, but then along the way up, all the mountain huts and the store with food at the top will be closed. You have to make sure you bring the necessary food and water for the trip up and down. Most likely there will be other post official season climbing, especially foreigners who don't listen to the official season dates (remember this is the country that officially closes beaches at the end of August, even though the weather is hot and summery!)
I just emphasize this: don't underestimate Mt. Fuji. It is famous as a tourist destination, but it is still a high and volatile mountain. It is not an afternoon stroll. The climb is very hard on your knees and you will be exhausted after the long descent through soft ash. During the day it can be brutally hot one moment and freezing the next. Make sure you have all the proper equipment: the ten essentials, rainwear, sunglasses, food, water, sun cream, a warm insulating layer, some form of shelter like a tarp or survival bag.
That said, every year thousands of completely unprepared people do the climb, so with a little preparation you should be fine. Have fun!
PS: If by chance it is not possible to climb Mt. Fuji I would suggest making an attempt at Kitadake, just west of Mt. Fuji. It is the second highest mountain in Japan and there is no closing date, though too late in the year (usually about the end of October) and bus service stops. It is a bit more serious than Mt. Fuji, but a lot more interesting and varied. You will have a wonderful view of Mt. Fuji from the top. There are quite a number of mountain huts along the way, so you will only need to bring basic shelter for emergencies. You can do a two-day, there and back trip, or a six-day, much more involved traverse of the range. To get there take the Azusa Limited Express from Shinjuku station to Kofu and from Kofu take a bus (or if you have the money, taxi) to Hirogawara. Climb from there. Pretty straightforward.