First off, when I was thinking about a trip that included going from Koyasan to the Kumano area a few years back, I found out that the busses from Koyasan to Tanabe do not run in the winter. You will need to check this, but I think this is because of snow.
That said, the JR 5 day Kansai Wide Area pass will help get you down to Shingu or KiiKatsurra from either Kyoto or Osaka. The one way fare from Shin-Osaka to Shingu is 7350 yen, which means the round trip is more than the cost of the pass. You then need to take busses in to the Kumano area.
I would place the JR pass at the beginning, or ending of your trip, because it will get you to the airport. I would also look into somewhere to visit in the extended area of the wide pass, such as Kobe, Himeji, Kinosaki Onsen, to name a few.
There is also the Ise-Kumano-Wakayama pass, which includes busses from Shingu and KiiKatsurra into the Kumano area. It is also a five day pass.
The JR pass will not help with Koyasan, and Kyoto for the most part. A bus day pass works well in Kyoto, and getting to Koyasan involves private trains.
You might think about either a Koyasan World Heritage Pass, or, a Kansai Thru pass. Just like the choice above, you will need to do some research, to figure out whatfs best for you.
The rest can be done with an IC (pay-as-you-go) card, or by just paying for a ticket.
If I was planning this, I would probably just stay my first night in the Namba Area of Osaka, and get the direct Nankai airport train to Namba. I would then go to Koyasan next, and use a Nankai World Heritage pass, to get there and back. To get from Namba to Kyoto, Ifd probably go to Nara by using Kintetsu trains.
I have already mentioned two private train lines, Nankai and Kintetsu. If you check the Kansai Thru pass itfs actually good on more that a dozen different private lines in the area. The thing is, if you are going from Namba to Nara, or, Nara to Kyoto, the cost is about 500-600 yen. If you plan to stop and stay the night, getting a Thru pass will not pay off. An IC card will end up costing less.
Itfs confusing, and you need to put in some time looking at your options. There is no magical pass.
I hope this helps as a starting point.
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