I'm flying into Tokyo Narita at the end of September and will be staying in Shinjuku. As I understand it taxis from the airport are very expensive.
What's the best way to get from the airport to Shinjuku?
Bear in mind that Shinjuku is a huge station and that it's quite easy to get lost or turned around in it. In light of that it may be helpful to figure out which exit you want to take in advance. Here's a map of the exists:
Also, there's an attached underground mall called Subnade (connected to the Shinjuku station via the East Exit, which takes you to the Metro Promenade, which, in turn, takes you to Subnade) with additional exits:
For my hotel the optimal exit was # 13 of Subnade. YMMV.
It's years since I did this but the airport limousine bus went straight to the hotel (Shinjuku Washington in my case). It looks like it still does.
This avoids the need to navigate Shinjuku station just after arriving in Japan, as well as the need to get from the station to the hotel with your luggage -- if the bus stops at your hotel (or the one next door).
As usual, Hyperdia will show you all available train connections, just be careful to choose "Narita Airport Terminal 1" (or 2) as your departure station and not just "Narita" (the latter is located in downtown Narita city).
The Narita Express is the only direct rail link between Narita airport and Shinjuku, but it is also the most expensive one. However, since cheaper alternatives require making at least one transfer at a central Tokyo station, I wouldn't recommend it unless you are a fairly seasoned user of Japanese urban rail (and if you were, you wouldn't be asking this question). Nevertheless, for your reference the cheapest train options can be found by unticking the "Limited Express" box in Hyperdia. This will exclude the more expensive limited express trains, such as the Narita Express and the Skyliner, from the results and keep only the cheaper, ordinary commuter trains.
As pointed out by Chris H, limousine buses are also an option well worth considering, especially if they stop at or near your hotel.
Access Narita limousine bus is the cheapest option to/from Narita airport. It costs only 1000 yen one-way to Tokyo or Ginza stations and preschool kids are free if sitting in your lap. So, for the family with two preschoolers it costs only 2000 yen one way. From Tokyo or Ginza, it's an additional 200 yen by train to Shinjuku.
Note however that it is only marginally (30 yen) cheaper than an ordinary train on the Keisei Main Line to Nippori. (From Nippori to Shinjuku is also an additional 200 yen.)
I am going there in April 2016,
Upon researching, I find that Limousine Bus can be pricier option.
Option 1: Access Narita bus from Narita -> Tokyo station and then JR train. (1500 yen)
Option 2: Keisei Limited Express from Narita -> Tokyo station and then JR train. (1200 yen)
Option 3: NEX Train for foreigner -> 2000 yen
Option 4: Limousine Bus -> 3100 yen
You can also check this site: http://www.japan-guide.com/e/e2027.html