The entrance exam is quite difficult. Kyoto university actually recommends applicants for the Phd program to first do a mathematics master at Kyoto University. Note there are certain scholarships which allow you to waver the entrance exam. I am not sure but I think it is this one -> http://www.studyjapan.go.jp/en/toj/toj0302e.html

Kyoto also offers cultural study programs called KUINEP. If you are looking for a gap year you can maybe try that.

For a math phd the odds are of course greater if you go for a lesser institute (Kyoto and Tokyo university are the top 2 universities of Japan). If you want more info hit me up on Discord under the same name.

I never met him. I think it was before the hype that I went to Kyoto. I was also in a different research group.