I'm pretty sure the railroad line down to Shingu is not JR. The pass may be no good on that trip. However, those tickets were not that expensive. It's a slow local - about 4 hours from Kyoto to Shingu.
If I remember this correctly, JR West manages the Kiseihonsen train route down the Kii Peninsula, from Tennōji in Ōsaka all the way back up to Nagoya. I would check the JR West site to be sure, but the relevant passes should be valid to Shingū. Although your estimation of time is about right, that four hour trip is actually a limited express - the local is slower than that. That just goes to show you how slow it can be to wend your way along the coast in Wakayama by train (it gets even slower once you pass Kushimoto to head north).
And just because I lived a couple of hundred meters from Kamikura (and had this number drilled in my head by neighbors proud of our mutual proximity to the shrine), I feel as though I should share that although it may feel like 650, there are 538 steps up to the jinja. But as most of those are set at about a 40/45 degree angle, it feels more like a thousand. Anno Sensei would lead classes at the top of the shrine each year during kangeiko when I lived there, and some of us were stupid enough to run up for exercise otherwise. If anyone is planning on visiting Kamikura, I would strongly recommend doing so on February 6, as this is when the annual O-tō-matsuri (fire festival) is held. It's great fun to see, and even more fun to participate in if you're able. Anyhow, it would be great to hear from anyone who lives there now (or is even just passing through) to see if kangeiko is still held on the grounds of Kamikura.
As for "real Japan," it would seem that it exists either contiguous with Brigadoon and Shambhala, in the fertile and misguided imaginations of anime fans, or as a product of the marketing offices of the Japan Travel Bureau. Caveat emptor.
btw: Howdy, Ron.