No, the JR Rail Pass would not be worth the cost for this trip. The only long distance travel you have is the round trip Tokyo-Kyoto-Tokyo. I'd just buy tickets for the (faster, more frequent) Nozomi shinkansen and forget about the Hikari green car. Obtain local prepaid transit cards - Suica or Pasmo - for local subways and such in Tokyo.

The only thing that might change the above would be if you are arriving and departing at NRT rather than HND. In that case, you could add in the cost of transportation to/from the airport, which would then make the Rail Pass pencil out. The cost of NRT-Tokyo (on the Narita Express) plus Tokyo-Kyoto-Tokyo plus Tokyo-NRT would exceed the cost of the Rail Pass and make the Pass worth the cost.

The itinerary seems a bit crowded. Some days, you have a number of sites that are not close together. If you spend much time exploring and enjoying the individual locations, you might have to trim your schedule. It is nice to have a lot planned but be prepared to edit on the fly.

Understand that June is the rainy season in Japan. You may have to be prepared to adjust your plans as you go to deal with weather. Some sites are not so good in the rain. I'd plan a fall-back in Tokyo in case of weather - perhaps a few hours at the Edo-Tokyo Museum in Ryogoku would be a nice and informative way to wait out the rain. Also, if you get caught out in the rain, remember that the ubiquitous convenience stores sell very inexpensive, almost throw-away, plastic umbrellas.

I NEVER plan a specific restaurant for lunch. Days become too structured that way. Since there are nice local places for lunch virtually everywhere in Tokyo and Kyoto, I prefer to just find a place when I get hungry rather than force the pace to arrive somewhere specific. I usually do the same for dinner but do understand your desire to dine in a specific way. To each his own.