Osaka to Tokyo incl. Koyasan, Nara & Takayama

Hi, this is my first trip to Japan with the following itinerary -

12 Oct Arrive at Osaka Kansai International Airport → Koyasan

13 Oct Koyasan → Nara

17 Oct Takayama → Kyoto

20 Oct Kyoto → Tokyo

23 Oct Tokyo → Narita

Please advise the type of rail pass I should buy or should I buy the train as I go?
Thank you.