The Japan Rail Pass and the JR pass are the same ticket (Japan Rail = "JR"), but I can see how that can be confusing! The Japan Rail Pass (JR Pass) can only be purchased outside Japan, and you can find our prices here.
I think for a short trip such as this I would aim for something like this to split your trip with a tour around Kyoto:
- 30th. Haneda > Tokyo
- 31st. Tokyo
- 1st. Tokyo > Kyoto
- 2nd. Kyoto (maybe Nara)
- 3rd. Kyoto > Tokyo
- 4th. Tokyo
- 5th. Tokyo > Haneda
This way you can recover from your flight with some sightseeing in Tokyo, and then travel down to Kyoto for a few days midweek. Finally, you can return to Tokyo for the remainder of the trip.
Rail cost wise, a 7 day rail pass will be the same cost as the return trip to Kyoto from Tokyo on the Bullet train. Of couse, the Rail Pass also means you can use it to do other day trip as you need, for example to Nikko from Tokyo, and you are able to use the JR Pass around Tokyo JR lines (but not the metro), however it is a borderline decision whether you will make good use of a rail pass or not.
If you do decide to travel to Kyoto by bus on arrival, then a 7 day rail pass will not be the most economical choice for you, however the trip to Kyoto will be pretty tough after a long flight so I certainly recommend considering staying over in Tokyo for a few nights on arrival.
Hope this helps!