Shinjuku Apartment trip
How to go to Haneda airport:
1. Métro de Akebonobashi jusqu'à Shinjuku (5 min) - 170 yen
2. JR train (Ligne Yamanote) jusqu'à Hamamatsucho (25 min) - gratuit pour les étrangers qui ont le JR Pass
3. Tokyo Monorail Train jusqu'à l'aéroport (20 min) - 470 yen
J'ai pris le métro de 5h10 à Akebonobashi et je suis arrivé à l'aéroport à 6h10.
Très simple et très rapide, même chargé...
1 From Akebonobashi to Shinjuku by subway. 5 minutes, 170 yen
2 JR Yamanote line to Hamamatsucho, 25 minutes, free of charge if you have JR Pass.
3 Tokyo Monorail to Haneda airport 20 minutes, 470 yen
If you don't change train:
Take a taxi to Shinjuku station (around 980 yen).
Take the Limousine Bus from Shinjuku West Gate directly for Haneda Airport. (1200 yen)
Sleeping in the airport for a couple of hours:
If your flight is 7:00 in the morning, you have to be at the airport at 5:00 a.m..
In this case recommending the last limousine bus leaving at 1:00 a.m. from Shinjuku
station to Haneda airport.You can spend few hours at the airport sleeping on the sofa.
It will be more economical way than spending one night in a hotel at the airport.
How to go to Narita Airport:
Take Narita Express or Limousine Bus from Shinjuku station. (3,000 yen)

Thanks for the information to Monsieur Julien Vandebeek-san.
e-mail to Shinjuku Apartment