Shinjuku Apartment trip
How to come from and go to Haneda airport:
Take a limousine bus from Haneda airport to Shinjuku station that will take you directly.
Take a taxi for ten minutes or take a subway for two stations to Akebonboashi. 
Comment on y va:
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 want to transit:
Take a taxi to Shinjuku station (around 980 yen) then,
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:/Narita Airport? It is the easiest way.
成田:Take Narita Express or Limousine Bus from Shinjuku station: 3,000 yen
羽田:Haneda Airpot :1,200 yen
 How to go to Disneyland?  
800 yen from Shinjuku station south gate by direct autobus.
400 yen for a child.
It is better than taking subway that requires transit two times because
you can just sleep or enjoy sightseeing relaxed.
 
Thanks for the information to Monsieur Julien Vandebeek-san.
e-mail to Shinjuku Apartment