Best way from Narita to Shimbashi station?

My family is arriving Narita on Jul 4, 2018. What is the best way to travel from Narita to Shimbashi station? Is the transfer at Nippori using skyliner easier than the transfer at Tokyo station using N'EX? Is there a roundtrip ticket option with either skyliner or N'EX that will give us a discount?

Thank you in advance for your advice!

How about Limousine bus?

There is no objective "best"; it depends on what your priorities are (price, speed, comfort, etc.).

check their offical site for list of available discount if any.

The change at Nippori station is a short walk; at Tokyo Station; the walk from the Narita Express platform to the Yamanote Line(and similar) platforms is much further and a sequence of escalators. All very doable but if this is a concern, changing at Nippori is probably easier.

Have you confirmed that you cannot choose Haneda Airport [HND]?

the walk from the Narita Express platform to the Yamanote Line(and similar) platforms is much further and a sequence of escalators.

You never need a transfer to the Yamanote Line in Tokyo Station,
unless the operation of the Yokosuka Line from Tokyo happens to be suspended or delayed much.

In Tokyo Station,
getting off a Narita Express train from the airport,
you usually can catch a Yokosuka Line Local train
departing at the same track or the other track beside the same platform (#),
which makes a stop at the next Shinbashi Station.
Also the Keisei-JR transfer in Nippori Station(s) is simple when you go through the transfer gate.

(#) Narita Express trains arrive and depart
at Sobu Underground track on the 5th basement of Tokyo Station,
where the Sobu Line (Rapid service branch) and the Yokosuka Line are jointed;
transfers to or from the other lines take more time.

To Shimbashi Station,
Tokyo Station is closer than Nippori Station.
Trains for Shimbashi are more frequently available
for the Yamanote Line section from Nippori
than for the Yokosuka Line section from Tokyo.

My family is arriving Narita on Jul 4, 2018

Which terminal of Narita Airport [NRT] will you arrive at?
(Narita Airport has three terminals.)
When in the day is your flight arrival scheduled?

What is the best way to travel from Narita to Shimbashi station?

Where exactly is your first destination?
Actually there are four Shinbashi Stations
: of JR, of Toei Subway, of Tokyo Metro, of Yurikamome.

Will there be a kid among you travelers?
A transportation cost shown below is for one adult passenger.

From NRT Terminal 1, 2 or 3
you can go first by low cost bus to Ginza
then by Tokyo Metro Ginza Line from Ginza to Shinbashi.
When no one carries big luggage or feels bad at a 1-and-a-half hour bus ride, this option may be suitable.

- The Access Narita bus
http://accessnarita.jp/en/home/#timetable-title-3386
: You pay 1000 yen for the bus ride to Ginza
and about 170 yen for a Tokyo Metro train ride from Ginza to Shimbashi.

From NRT Terminal 1 or 2
you can take one or two trains
to Shinbashi Station of JR or of Toei Subway.

- JR-East: N'EX TOKYO Round Trip Ticket
https://www.jreast.co.jp/e/pass/nex_round.html
: You pay 2000 yen per way
to Shimbashi or back to the airport.
: The ticket from the airport allows you
to change trains inside JR gates in Tokyo Station then go to Shimbashi.

- Keisei Skyliner & Tokyo Subway Ticket
http://www.tokyometro.jp/lang_en/ticket/value/airport_bus/index.html#a...
: You pay 2000 yen / 3900 yen
on the one-way / the pair of round-trip Skyliner ticket(s)
and about 170 yen for a JR train ride from Nippori to Shimbashi.

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gSobu Underground tracksh,
gShinbashih should be
gShimbashih
and
gto Shimbashi or back to the airporth should be
gto Shimbashi and back to the airporth
in my previous response.

Thanks.

Firas:
it depends on what your priorities are (price, speed, comfort, etc.).

Yes!

Some prefer railway to bus, some the reverse.
A backpacker may not mind a ride without seat reservation.
A good walker may be ready to use a staircase in a station crowded with passengers.

Keisei Skyliner & Tokyo Subway Ticket

You can purchase Tokyo Subway Ticket in NRT before heading for Tokyo
then make it cover the Tokyo Metro fare from Ginza (G-09) to (G-08) Shimbashi.

When you get to Shimbashi in the night,
it may be better to pay the regular fare
and start to use the subway pass in the morning of a later day.

- Tokyo Subway Route Map
https://www.tokyometro.jp/lang_en/station/rosen_en_1803.pdf

Thank you to all for your great suggestions! Very much appreciated!