I am sorry l can't answer your question but I like the Limousine bus from Narita that goes to the door of the better hotels and costs 1,800 yen. One of the baggage boys takes your bags from the trolley and you don't touch them again until you get to your hotel room. I used to use the Narita Express, the slowest train in Japan, which is usually takes longer door to door and is a lot more hassle than the bus. Labour is expensive in Japan. A taxi will cost about 250 US and l doubt if a private transfer will be any cheaper. It is no faster than the bus.
You realize the distance from the airport to hotel (Narita for Tokyo and Kansai for Osaka) is around 80km (depending where you stay) and a taxi is going to cost you around US$200. I had a guest a few years ago spent US$600 on a round trip to Narita to pick up his daughter (he didn't realise it was so much), and another that wanted a chauffeured car out to Narita and he dropped US$600 on the one-way trip.
Kyoto to Koyasan is around 130 km and 2 1/2 to 3 hours. Last private vehicle we hired in Kyoto cost 6,000 yen/hour, so you would be paying for their round trip or around US$500 minimum.
Keep in mind that Japan actually has a well-functioning public transportation system. English signage on the train system is excellent. Using japan-guide.com, hyperdia.com, and google maps, you can find affordable public transportation for all the places you mention.
For Koyasan, try https://www.japan-guide.com/e/e4904.html?aFROM=2158_4900
For the airport trips, go to https://www.japan-guide.com/e/e627.html and look under Airplanes for the airport you want.