Mt Fuji is clearly one of the most iconic summits on Earth. It is climbed by thousands of people each year, and there is a race to the summit. There are several popular routes, which are described here: http://www.fujisan-climb.jp/en/trails/index.html
There is an annual ascent race, the Fuji Mountain Race, from Fujiyoshida City Hall to the top, and we do not track records for the event. Toru Miyahara has that record in 2h27m41s in 2011, for the women Yoshimi Hoshino ran 2h36m16s in 2001.