If you give them the flight number you are covered, even in the wee hours. They will upgrade you free if the car class you reserved is gone by the time you arrive. A rental "day" is always 24 hours; the clock starts running when they print the paperwork (or in the case of Natl's EA, when you arrive at the booth.)
The only time the arrival time really can trip you up is when it is near the cut-off for a given rate restriction. For instance; if you have a weekend rate, you almost always are required to pick it up no earlier in the week than noon on Thursday. If you showed up and wanted to pick your car up earlier, you would have to pay a higher rate. Drop off too late in the week, and the same thing would happen.