The Short Answer
A standard private Mount Fuji day tour from Tokyo (5th Station, a lake viewpoint) runs surprisingly cheap at $45 flat for 1-4 people on some operators' basic packages, while a fuller private tour with an English-speaking driver and guide for the whole day runs $330-420 flat for 1-4 people. A combined Mount Fuji and Hakone private day runs $1,000+ flat for a standard car. A multi-day private climb (the 2-day Prince Route) prices per person from the start, at roughly $1,700+ for up to 5 climbers, since mountain-guide certification and safety equipment are genuine per-head costs.

From Tokyo: Mount Fuji & Hakone Private Day Tour
A real example of flat pricing for 1-4 people on a fuller private day tour.
From $332
Full day
