Best route to Shinjuku

Am arriving at Narita airport on a Saturday afternoon end of Marcj and staying at the Sunroute Higashi Hotel in Shinjuku after a long flight I want the quickest route to my hotel. Is the best option the Narita Express train to Shinjuku station then a taxi from the station to the hotel? Thanks in advance.

Limousine from Narita airport to Shinjuku station West exit

Time table
https://www.limousinebus.co.jp/en/platform_searches/index/2/212

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From Shinjuku Nishiguchi station (Shinjuku West station)@Toei Oedo line to Higashi Shinjuku station
2 min

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From Exit B2@Higashi Shinjuku station to Hotel Sunroute Higashi Shinjuku
About 1 min walk

Agree on the limo bus. Easiest way to Shinjuku. Not too pricy either...about 3000 yen I believe.

The easier way. Take the Keisei railway's Skyliner to the terminus at Ueno. It takes about 40 minutes.
When you exit the station turn right and walk about 3 blocks to the subway station (Ueno Okachmachi). Take the Oedo line 6 stops (12 minutes) to Higashi Shinjuku.

Thank you very much for the info. I now have a Pasmo card and with the info provided am feeling more confident.

I am a fan of the Keisei Skyliner, and I have taken it two of the three times that I have gone from Narita Airport to Tokyo, but I believe that the JR Narita Express has a direct train to Shinjuku. I believe that a direct train is more convenient than switching trains.

If you do take the Narita Express make sure that you check with the ticketing agent. After the Narita Express hits Tokyo Station, some of the Narita Express trains go towards Yokohama, and some go towards Shinjuku.

The Limo bus is also a good option.

If you do take the Narita Express make sure that you check with the ticketing agent. After the Narita Express hits Tokyo Station, some of the Narita Express trains go towards Yokohama, and some go towards Shinjuku.

There's no need to worry about this, as the NeX is all assigned seats. If you buy a ticket to Shinjuku, and you sit in your assigned seat, you will go to Shinjuku, not Yokohama.

The Keisei + Oedo is a clever route, but NeX or limo bus plus taxi might be more relaxing if you're tired after a long flight.

I'm tempted by the taxi once I get off the Limousine Bus - but read that taxi drivers don't speak English? Is it easy enough to just show picture/name of hotel? As only one stop on the metro I assume its not far?

I think it easier to catch a limousine bus for Ginza, Shiodome direction and then get off at "Park Hotel Tokyo". "Shiodome station" is right opposite the hotel and get in Toei-Oedo line from there for Higashi-Shinjuku station(17 stations, 270 yen).
BUT, the limousine bus for Ginza, Shiodome area is limited.
http://www.limousinebus.co.jp/en/platform_searches/index/2/71

Is it easy enough to just show picture/name of hotel?

I believe so. You can write in advance "Hotel Sunroute Higashi Shinjuku" down on a piece of paper and show it. That may well be enough for a taxi driver waiting around the Shinjuku railroad terminal, I suppose, even if he/she does not speak English.

A taxi route from the limousine bus stop near Shinjuku Station to the hotel is estimated to be less than 2 kilometers long. When a taxi runs a smoothly, a one-way ride to the hotel may last for 5 minutes or so and require 730 yen or 820 yen per car. I recommend you will prepare some 1000-yen banknotes and 100-yen coins in case your PASMO Card cannot be accepted.

You can check on this website the location of a limousine bus ticket counter in your NRT terminal.
http://www.narita-airport.jp/en/guide/service/list/svc_32.html

Different preferences for different people. I dislike the airport-limo bus (used it a few times) and the N'Ex train does go all the way to Shinjuku for about the same price.