Handling a very early flight from Tokyo with public transport constraints

Our flight leaves Tokyo very early, before I'd normally think about getting out of bed. We're staying somewhere convenient for sightseeing, not necessarily for the airport. I'm checking first-train times rather than assuming public transport runs all night. If the rail connection can't get us there comfortably before check-in closes, would you move to an airport hotel the night before or arrange a fixed-price car? I don't want a cheap ticket to become an expensive scramble.