A limited express bus runs 6 round trips a day from Osaka and 3 round trips a day from Kobe (Sannomiya).*There are no highway bus services from Kyoto City. The best way to catch the bus would to be to travel to Osaka and get the bus from there.
More Information Japan Bus Lines
If you are looking for transportation in Japan, JAPAN BUS LINES has an excellent team with bus companies from every region in the country, and a leading highway bus network covering many cities. You can make reservation tickets between Osaka-Kobe and Kinosaki Onsen.