Hi there,
The JR Pass will work very well well with your itinerary. There are some parts however that can cause confusion, think of Hakone and Koya-san.
Starting with Hakone, the Hakone Area can be reached by Shinkansen from Tokyo. The Shinkansen station is called Odawara and takes 36 minutes to reach. From there local (non-JR) transport runs into the city. For travel within hakone
Koya-san also requires a bit of preparation to travel to. Travel between Shin-Imamiya station (in Osaka) and Koya-san is not covered by the JR Pass, as this route is travelled on the Nankai Express (a private railway company). To go into detail, take a local JR line to Shin-Imamiya station and transfer to the Nankai Express. You can buy a ticket at Shin-Imamiya station for all the way to Koya-san. The Nankai station for Koya-san is called: Gokuraku-bashi from where the Koya ropeway takes you up the mountain. You can also buy a Koya Heritage ticket that covers this route and gives you above 20% in savings, compared to purchasing normal tickets.
The JR Pass covers the rest of your travels and the station names are the same as the city names, for instance, Kyoto is Kyoto station.
Hope this helps,
Daniel
