A 14 day JR pass will not be worth it. It's more expensive than individual fares for your travel. Forget about that.

For the trip from the airport to Shinjuku, you could take a Limousine Bus - door to door service from the airport to the hotel:
Park Hyatt Tokyo?Routes,Timetables,Fares?Airport Limousine Bus

If you dislike buses you could take the Narita Express train non-stop service to Shinjuku Station and catch a cab from there. (And if you want to get there sooner, you could take the Keisei Skyliner to Nippori, then change to the commuter train (Yamanote Line) as far as Shinjuku. But you might prefer not to deal with a change of trains and a ride on a commuter train with luggage after a long flight ...).

To get around Tokyo, I recommend that you get a PASMO card - a stored value contactless card that works on buses and trains (and some taxis). You can get the balance on a PASMO card refunded at Haneda airport at the end of your trip. (The alternative to PASMO is SUICA, but I don't think you can get a refund on that at Haneda).

If you wish, you could use a 7-day pass from the 9th to the 15th. That's exactly 7 days. You could use it for the Tokyo-Kyoto round-trip (27K yen), the Kyoto-Osaka round-trip (2K yen), the Kyoto-Nara round-trip (3K yen), and the monorail journey from Hamamatsucho to HND (500 yen)**. The pass costs around 29,000, so you're looking at savings of around $35. You might also appreciate the freedom to take other trips. Personally, I would rather have the freedom to ride the more frequent Nozomi trains, so I would do without a JR pass for this trip.

** JR pass is valid on the monorail, but to be honest, I would prefer to take the Keikyu line from Shinagawa to HND, rather than dealing with the connections (take Shinkansen to Shinagawa, followed by Yamanote line to Hamamatsucho, followed by Monorail to HND). JR pass is not valid on Keikyu line, so it's better to pay the fare (with your PASMO card) IMHO.

When you are in Kyoto, you may wish to use the one-day bus pass - 500 yen - to get around town.
https://www2.city.kyoto.lg.jp/kotsu/..._card_bus.html

In Nara, if the weather is nice, bicycle rental is a good way to get around (in Kyoto too, actually)
http://nara-rent-a-cycle.com/en/