>>480047Run three times a week for 15 minutes, try to run as fast as you can, measure your time and distance and improve, this is fun and will keep you going. Also do squats and some other exercise 3 times a week for 15 minutes. Don't go run up the stairs for 45 minutes like
>>480057 suggests that's super boring and will loose your appetite for it. Try to go once every 2 weeks on a hike(20 km) on a free day in some nice patch of nature close to where you live, if this gets too easy just take some ballast in your backpack or do more distance. The last two weeks before you do the climb are important, try to train as much as possible, best is hiking in the mountains, but if you don't have that possibility fill the empty spots in your schedule with exercises and running, be careful not to get an injury though. The two days before the climb you try to rest.
Also look into what you have to eat, during preparation but also during the climb, i don't know if you need this information or you already know what you should be eating, you can also ask /fit/. Also get good boots, if you need info ask here or check one of the other boot threads.
Mount fuji is only 3.776 m high. If you're skinny and prepare yourself it will be like a walk in the park and you'll enjoy it a lot more. You might get a new hobby and getting in shape never hurts. Good luck!