When in the day is your flight scheduled?
Actually your hotel is a little apart from (JR) Shinjuku Station.
The station which is closest to your hotel is Seibu Shinjuku Station.
You can travel from (JR) Nippori Station to Seibu Shinjuku Station by the following route,
in around 25 minutes, at 320 yen per adult passenger.
Nippori Station
| JR Yamanote Line's Inner Track
| (for Ikebukuro and Shinjuku)
Takadanobaba Station
: Pass through JR-Seibu transfer gate
https://www.jreast.co.jp/e/stations/e938.html | Seibu Shinjuku Line
Seibu Shinjuku Station.@
Shinjuku Station of JR is huge and complicated.
https://www.jreast.co.jp/e/stations/e866.htmlA Narita Express ride to Shinjuku Station may seem perfectly nice, but I would point out that the Narita Express platform (for Tracks 5 and 6) is on the further side both to your hotel and to your bus terminal (in front of Keio Department Store).
To avoid walking (with your luggage) for a while on the underground concourse, you can take the fowling route to the taxi terminal on the 3rd floor of Busta Shinjuku.
A one-way taxi fare to get to your hotel is estimated at around 900 yen per car (, i.e. around 450 yen per person, because you are two travelers).
: 1. Go up by escalator to the bridge for Koshu-kaido Gate and New South Gate.
: 2. Pass through Koshu-kaido Gate.
: 3. Turn to the left.
: 4. Walk a little to the elevator or the escalator leading to Busta Shinjuku.
: 5. Go up to the 3rd floor, where the taxi terminal is.