That's pretty annoying that the hotel wants to charge a whole night. I've been allowed to check into hotels in early morning in US, HKG, etc. Hotels in Japan are notorious for their lack of flexibility with extending check-out time. But I don't see why they should care about early check-in if the room is available already.

Anyways, there's a capsule hotel type place (actually more like tiny bedrooms than capsules) called First Cabin at HND where they have "short stay" option in which you pay for the room by the hour. I just checked their website, though, and it says the period of 7p-10a is for the overnight use only. So I'm not sure if that's an option for you. And it's in the domestic terminal.