Mount Nantai is a sacred mountain owned by Futarasan Shrine's Chūgushi. The path is open between 5 May and 25 October, and there is a 500¥ fee to climb the mountain.
If you want to know more about hiking Nantai San, there is plenty of useful blogs which give valuable information on how to get there, what is the best time to climb the mountain, how you should prepare, etc. Do your research! Please, note that you can also contact me by PM and it will be a pleasure to answer your questions from my experiences.
Here's a few images I took last year during my photo excursion to Nantai San's summit in Nikko.