New South Gate is open from 5:45 am;
it is closed for the earliest hour when some trains are already running.
When you take a taxi to Busta Shinjuku,
which is a large complex of expressway bus and taxi terminals,
MIRAINA TOWER Gate sounds best.
In Busta Shinjuku,
the taxi arrival space is around the northeast corner of the floor,
closer to MIRAINA TOWER Gate
than to Koshu-kaido Gate and New South Gate.
- Map of the 3rd floor of Busta Shinjuku
http://shinjuku-busterminal.co.jp/en/floormap/3f.html
When your taxi goes smoothly from Yotsuya-Sanchome Station,
you get to Busta Shinjuku in about 5 minutes,
and the taxi fare is estimated at around 800 yen per car.
You may need 5 to 10 minutes more to get to the Narita Express platform.
(1) Your taxi arrives to Busta Shinjuku, on the 3rd floor.
(2) Walking along the taxi lane, you turn to the right,
then pass by a few elevators seen on your left.
(3) You turn around counterclockwise to the escalator.
(4) You go down by escalator to the 2nd floor.
(5) You turn around clockwise,
then walk until you see MIRAINA TOWER Gate on your right.
(6) You turn to the right, then pass through the gate.
(7) You go down by elevator or escalator
to the platform for Tracks 5 and 6.
(8) Your Narita Express train (for NRT) departs at Track 5.
- Map of (JR) Shinjuku Station
https://www.jreast.co.jp/e/stations/e866.html : In this map, the north is about the right.