Florabee, I also did only have one hour to catch our bus to Ikebukuro area in May, so what we did was we reduced our check-in baggage, and we paid for seats near the plane's exit. That way you would be slightly ahead of other passengers to board the bus to the terminal building and clear immigration while your bags are being delivered. The airbus ticketing counter is very noticeable when you come out of the arrival gate, again you need to line up for your turn to be served.
Your back-up plan should be to stay in Haneda area for the night unless you have something you have to do very early next morning, in our case we had to meet at 7.40 the next morning for a pre-booked tour.
Airport bus will charge night fare at double the normal rate for midnight or past-midnight travel. So the night fare to Shinjuku would be JPY2,000 per person. If you have 3 persons or more, you might want to consider a taxi taxi which charges a flat fare of JPY6,000 per car. (plus toll I think)