Shinjuku Washington Hotel to Narita Airport

I'm leaving for Japan in about two months, and I have a long layover in Tokyo and the next morning I depart for Kansai from Narita Airport.

I'll be staying at the Shinjuku Washington for a night, but my flight is at 7:25am the next morning. What's the fastest way for me to get to Narita Airport from Shinjuku station? And what time do the trains start running?

I strongly recommend to stay near the airport for an early-morning departure. If that is not an option, the fastest way to the airport would be the Skyliner 1, which leaves Nippori at 6:03 and arrives at the airport around 6:40. Departure from Shinjuku would be with the 5:35 Yamanote Line. That is enough time for a person familiar with the airport to catch the 7:25 flight, but it may be risky for a person unfamiliar with the airport.

There are even earlier train departures, but they are more time consuming. You can look the up here:
http://www.hyperdia.com/

Taking a taxi would be even faster, but it is prohibitively expensive:
http://www.japan-guide.com/e/e2027.html?aFROM=2027_3011