We have arrived in that time frame a number of times and we drive down from northern Maryland. Assuming that you don't have a room for that night (and maybe don't want to spend too much money)...
We have done online check-in and then when we arrive, we go to the main lobby and talk to the night person who is on duty just to let them know that we have arrived 'much earlier than expected' because we drove all through the night instead of stopping. Most of the time they will just tell you that you have to wait, but twice in our many trips they have said 'well, your room is ready and empty, so go ahead.' We have been given a room prior to the 6am 'start of day,' but it certainly doesn't happen very often.
We have also just dozed in the lobby and also dozed in the car in the parking lot of our resort. If you do want a hotel for that night, we've had a very good experience at the Holiday Inn right by Disney Springs - convenient, but not all that cheap. Near the airport are generally decent hotels and cheaper. Careful of I-drive hotels, though - we've had some real stinkers...