JR rail pass vs single ticket

We will visit in May . We arrive at Narita Airport and will stay in Tokyo for 2 nights. Need advice on transportation between airport to Akasaka Station.
Head to Hakone , stay for 1 night.
Head to Kyoto, stay for 4 nights. Plan a day trip to Nara. Head to Koyasan, stay for 1 night
Head back to Tokyo with a visit to Osaka, stay in Tokyo for 3 nights
Leave Tokyo via Haneda Airport.
Would like to have information and advice on transportation from Narita Airport to Akasaka Station, from Akasaka Station to Haneda Airport.
Would like to have advice on with my itinerary, will this be worthwhile for me to have JR rail pass or just have single ticket and Hakone pass.
Your expertise is much appreciated. Thank you.

A 7 day pass pays off for a round trip between Tokyo and Osaka.

For your travel between Narita Airport and Asakusa, I would suggest using the Viator.com private car pick up. That's what we are doing. After 18+ hours of travel, the last thing we want to do when we get to Japan for the first time is try to figure out train/subway travel. It seems pretty affordable - I think around $47/person.

We are also staying in Asakusa in May. Hope that helps!

There are also limo buses that go to some of the hotels in Akasaka (not asakusa) - https://www.limousinebus.co.jp/en/bus_services/narita/akasaka.html

I think as long as I read your itinerary, even if you buy only 7-days JR Pass, you would not be able to complete it.
JR Pass can use the JR line company, other transportation is not available. To use them, you will need to purchase the Pass of them in addition to the JR Pass.

Narita-Tokyo One of either:
Limousine & Metro Pass
http://www.limousinebus.co.jp/discount_ticket/en/index.html
KEISEI SKYLINER & METRO PASS
http://www.keisei.co.jp/keisei/tetudou/skyliner/us/special/

Hakone:
Hakone Free Pass
http://www.japan-guide.com/e/e2358_008.html
Since you have a JR Pass, you go to Odawara Station aboard the JR, you can buy it.

Kyoto:
One or two-day tickets for sightseeing in Kyoto
One-day passes for City Buses
http://www.pref.kyoto.jp/visitkyoto/en/info_required/transportation/bu...
https://www.city.kyoto.jp/koho/eng/access/transport.html

Nara:
One-Day Free Bus Pass for Foreigners
http://narashikanko.or.jp/en/
If you're going to Toshodaiji and Yakushiji, or Horyuji, can not be covered by this. It is necessary to purchase the Nara Nishinokyo Ikaruga Free Ticket.
Nara Nishinokyo Ikaruga Free Ticket
http://www.narakotsu.co.jp/rosen/free-ticket/ikaruga_free.html (Japanese)

Koyasan:
Koyasan Sekaiisan Kippu
http://www.nankaikoya.jp/en/stations/index.html
http://www.nankai.co.jp/traffic/otoku/koyasan.html (Japanese)

Osaka One of either:
Enjoy Eco Card
http://www.kotsu.city.osaka.lg.jp/general/eigyou/price/jyousyaken_anna... (Japanse)

Osaka Amazing Pass
http://www.osaka-info.jp/osp/en/index.html
Entrance fee of tourist facilities such as Umeda Sky Building and Osaka Castle is free.

Tokyo:
http://www.jnto.go.jp/eng/arrange/travel/practical/tonai_ticket.html

transportation from Narita Airport to Akasaka Station, from Akasaka Station to Haneda Airport.

This station is Akasaka Station on subway Chiyoda Line, right?
- Tokyo Subway Route Map
http://www.tokyometro.jp/en/subwaymap/pdf/routemap_en.pdf

(A) Route from NRT

This route requires about 1 and a half to 2 hours.
If you choose a Narita Express, a N�fEX Tokyo Direct Ticket (1500 yen) will pay off.
Seats of Narita Express trains require reservation.
The subway fare is 170 yen if you purchase a paper ticket.

Narita Airport / Airport Terminal 2
--[JR-East: "Narita Express" or Rapid]---
Tokyo --transfer-- Tokyo (M-17)
--[Tokyo Metro subway Marunouchi Line]--
(M-14) Kokkai-gijidomae (C-07)
--[Tokyo Metro subway Chiyoda Line]--
(C-06) Akasaka.

Transfer in Tokyo:
Sobu Underground Track 1 or 2 on B5F --two escalators up--
Marunouchi Underground Central Gate on B1F (JR) --pathway--
(subway) Gate --escalator down-- Track 1 on B2F.

Transfer in Kokkai-gijidomae:
Track 1 on B2F (M-14) --elevator down-- (C-07) Track 3 on B6F.
- Map of Tameike-sanno Sta. (G-06, N-06) and Kokkai-gijidomae Sta. (C-07, M-14)
http://www.tokyometro.jp/station/kokkai-gijidomae/yardmap/images/yardm...

(B) Route to HND

This route requires about 45 minutes to 1 hour.
The fares in total are 672 yen if you use an IC card (Suica, PASMO or so).

Akasaka (C-08)
--[Tokyo Metro subway Chiyoda Line]--
Track 6: (C-08) Kasumigaseki (H-06): Track 4
--[Tokyo Metro subway Hibiya Line]--
Track 4: (H-09) Higashi-ginza --transfer-- Higashi-ginza (A-11): Track 1
--[Toei subway Asakusa Line + Keikyu lines: direct train]--
(the 2nd last) Haneda Airport International Terminal.

- Map of Kasumigaseki Sta.
http://www.tokyometro.jp/station/kasumigaseki/yardmap/images/yardmap.g...
- Map of Higashi-ginza Sta.
http://www.tokyometro.jp/station/higashi-ginza/yardmap/images/yardmap....