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.
Paul Deckret submitted the sea-summit-sea variation. Any route can be used, but Paul recommends the Murayama Kodō pilgrimage trail (~76km, see also Paul's FKT report for GPS data):
This route begins at the ocean, not far from JR Yoshiwara Station in Shizuoka Prefecture, and first travels along asphalt roads through Fuji City to the Murayama Sengen Shrine. The route joins up with a proper, though sometimes poorly marked, hiking trail from a point just beyond the shrine. That trail continues along an ancient pilgrimage route known as the Murayama Kodō through the forest that flanks the lower slopes of Mt. Fuji before joining the Fujinomiya Climbing Route at the 6th Station. From that point onward the route simply follows the standard Fujinomiya Route up to the high point of Mt. Fuji, known as "Kengamine." The return to the ocean is a simple retracing of your steps back down the Fujinomiya Route to the 6th Station, back along the Murayama Kodō to the Murayama Sengen Shrine, and then on the roads through the Fuji City back to the same spot of ocean just beyond JR Yoshiwara Station. To make it more clear and avoid any confusion, I'm attaching a link to a custom Google Map I made from a combination of my GPS data on the route marked with the location and names (in Japanese) of the 18 landmarks that I believe should all be included in order to be an official FKT on this route.
I believe this is likely the quickest approach for a sea-to-summit of Mt. Fuji. The ancient Murayama pilgrimage trail is not very well known or used, and very little, if any information is available on it in English. But it is dotted with historical markers throughout, passes through beautiful verdant forest, avoids the roads as much as possible and is an enjoyable route worth knowing about in my opinion.