According to a student helper (Megumi Murakami) at the University of Tokyo, there are three ways to come to Shinjuku Park Inn from Narita Airport.
Airport Limousine to Shinjuku Station West Exit, and taxi.
Pros: Simple plan. Minimize the walking distance.
Cons: The bus ride from Narita to Shinjuku is subject to traffic condition of Tokyo. During rush hour, delay can be significant.
Airport Express (N'EX) to Tokyo Station, and subway.
Pros: N'EX is the express train service from Narita to key stations in Tokyo, which is more comfortable than the limousine bus and is not affected by traffic condition on the road.
Cons: You have to change the train at Tokyo Station, which can be very crowded, although the direction is clearly marked. Moving around the station with a large bag can be a challenge, although most stations are equipped with escalators and elevators.
Airport Express (N'EX) to Shinjuku Station, and taxi.
Pros: Reliable service of train and a short taxi ride to the hotel
Cons: You must ride a right "part" of N'EX to go to Shinjuku Station. Navigating through Shinjuku Station to a taxi stand is a challenge, as it is the busiest train station in the world.