N'EX is the express train connecting Narita airport to the metropolitan area of Tokyo. Some N’EX trains go to Shinjuku Station.
Here is the tricky part.
At Narita airport station, the N'EX train heading to Shinjuku Station can be a "subset" of a "longer" train. Usually, at Tokyo Station (which is the main stop of N'EX), the train is split into two shorter trains. One half goes to Shinjuku Station, while the other part goes to a different location, passing Shinjuku Station. It is important to identify the "right" part of the train when you get on the train at Narita Airport Station.
At Shinjuku Station, take a taxi to the hotel. (Show this to the taxi driver.) The fare should be around 1000 Yen, but should not exceed 2000 Yen. Shinjuku station is the busiest train station in the world. Finding a taxi stand might be a challenge. You may find this You Tube clip useful, even though it is quite old. We can learn from other's experience.