I used to live in Tokyo and can help a bit with some of the details on how to get there. I also climbed Fuji-san twice on the route you mentioned, but didn't do a ski descent. I think it is illegal now, but I am not positive about it. They were debating it while I was there because they got tired of rescuing people. Also, I started from the 5th station rather than the Shrine. I think it is rather far from the Shrine. The climbing season for Fuji is usually open for parts of July and August. I would recommend going then because the transportation system is running more frequently during this time. As a foreigner, your odds of successfully hitchhiking are slim. I don't know if it is legal or not.
You will most likely fly into Tokyo (Narita). You will want to take the Narita Express to Shinjuku. It costs about 3300 yen -- $30 US.
http://www.tokyoessentials.com/arriving.html
Do not take a cab -- it will cost you $300.
You will want to take the JR Chuo line (Orange trains -- track 9 or 10 I think in Shinjuku) from Shinjuku to Otsuki and then go to Fuji Yoshida. I think it is about 2 hours.
This should help...
http://www.hyperdia.com/cgi-english/
(Use Fujiyoshida as the destination)
Another option is to take a bus from Shinjuku. We did this both times. In the climbing season they have buses that go straight to the 5th Station.
Just a warning...Shinjuku is insane. It is probably the busiest train station in the world...ask for help, if you haven't been there or don't read Japanese you will need it.
Japan is a complicated nut to crack due to the language and culture differences. Regardless, it is a wonderful place to visit. Best of luck with your trip and let me know if you need anything else.
Cheers
Todd
PS -- For what it is worth, if I was going to fly across the Pacific to see beautiful mountains in Japan I would go to Kamikochi to hike in the Japanese Alps. Mt Yari, the 3rd highest peak in Japan, is fantastic. If you want to climb and ski a volcano go to Sapporo, catch a bus to Niseko (the Japanese refer to this place as Powder Heaven...which it is, the snow is better than the fluffy stuff they have in Utah), and climb Mt Yotei and ski down.