You have various options to get to Shimbashi Station(s); I list below three of them.
: The first option is what John recommends.
: The transfer in the second option is quickest.
: The cost on the third option is lowest.
[Keisei Skyliner + JR Local train]
A Keisei-JR transfer in Nippori Stations is relatively short and easy.
There is a transfer gate between the two stations.
You can avoid staircases on the transfer route, using escalators or elevators.
Please make sure to choose in Nippori Station the correct track of the Yamanote Line to get to Shimbashi, which is Track 10 on its outer loop.
- JR-East: Map of Nippori Station
http://www.jreast.co.jp/e/stations/e1184.htmlKeisei Skyliner requires a surcharge, and its seats should be reserved.
The total of the basic fare and the Skyliner surcharge is about 2500 yen per way / about 5000 yen per round trip per adult passenger.
If you plan to visit many sites in or around Tokyo which can be accessible mainly by subway, a package of a (pair of) discount Keisei Skyliner ticket(s) and a Tokyo Subway Ticket may be a good deal.
- Tokyo Metro: Discount tickets for customers using Narita Airport
http://www.tokyometro.jp/en/ticket/value/airport_bus/index.html#anc02: A (pair of) Skyliner ticket(s) in this package gives you a discount of about 500 yen per way / about 1000 yen per round trip.
: A Tokyo Subway Ticket allows you to have unlimited train rides on Tokyo Metro and Toei Subway lines within 24, 48 or 72 hours.
- Keisei Electric Railway: Timetable for trains leaving Narita Airport
http://www.keisei.co.jp/keisei/tetudou/skyliner/us/timetable/index.php[JR Narita Express + JR Local train]
A JR Narita Express train brings you to Tokyo Station, then you can take a Local train on the Yokosuka Line for a few minutes to next Shimbashi Station.
Usually this transfer is quickest because the Narita Express train and the Local train use either the same track or the two tracks beside the same platform.
The tracks and the platform are in the Sobu Underground annex of Tokyo Station.
After arriving to JR Shimbashi Station, it may be better to get through Karasumori Gate and go out via Karasumori Exit, which are both on the ground floor.
The walking route from the nearest Shiodome Gate includes a staircase to go up, which leads to Shiodome Exit on the ground.
- JR-East: Map of Shimbashi Station
http://www.jreast.co.jp/e/stations/e877.htmlJR Narita Express requires a surcharge, and its seats can be reserved.
If you plan to get back to NRT within 14 calendar days, a NfEX TOKYO Round Trip Ticket package may be a good deal, which is sold only to non-Japanese passport holders.
- JR-East: a NfEX TOKYO Round Trip Ticket
http://www.jreast.co.jp/e/pass/nex_round.html: With a pair of tickets in this package, you can travel at 4000 yen per adult passenger, within 14 calendar days, from Narita Airport Station or Airport Terminal 2 Station to a JR station in the designated area then back, mainly by Narita Express.
: The station until which you use the first ticket and the station from which you use the second ticket can be different.
- JR-East: Timetable for Narita Express trains leaving Narita Airport
https://www.eki-net.com/pc/jreast-shinkansen-reservation/English/wb/co...[Access Narita bus + Tokyo Metro Local train]
You can use the Access Narita bus to Ginza then take a Local train on the Tokyo Metro Ginza Line from Ginza Station (G-09) to next Shimbashi Station (G-08).
The cost is less than 1500 yen per adult passenger.
A possible demerit of this option is that your route in Tokyo Metro Ginza Station to the Ginza Line platform is longer when you avoid staircases than the route which includes them.
- Tokyo Metro: Map of Ginza Station (in Japanese)
http://www.tokyometro.jp/station/yardmap_img/_station_%E9%8A%80%E5%BA%...: In this map, you see an elevator near the staircase leading to Exit C5. The Access Narita bus stop is around the entrance of this elevator on the ground.
- The Access Narita: Timetable for buses leaving from Narita Airport for Ginza
http://accessnarita.jp/en/home/#timetable-title-3195: Concerning a ride from NRT Terminal 1 or 2, seats are secured on the first-come, first-served basis.
: When you take a bus from NRT Terminal 3, you can try to book your ride at the ticket counter.