Not really. Most are nominally private (although both the big Tokyo subways aren't - Toei, the clue is in the name), but have both a large proportion of their shares owned by governments (local and national), and deep ties to a large conglomerate (keiretsu) that in turn owns a lot of businesses in the relevant area and has deep ties into the political system.

Yeah, the way "private"-ly owned infrastructure here is probably rather different than just selling stuff to foreign companies. I was just reading about the private toll roads in Japan, but it seems they're only operated and maintained by private companies, while the ownership is retained by corporations that are owned by the government (sort of like MTA in NYC).