As others have noted, if you want a room ready when you get there at ~5am book from the night before and make sure to note your arrival time and that you booked starting the night before. I haven't had a problem but I know that others have run into issues where they may not automatically get credit for the first night and need to follow-up afterwards.

The thing that hasn’t been mentioned and should be obvious is that even if you call the hotel and you get an appreciative response saying that they’ll let you check-in early, the reality is that if the hotel is booked solid the prior night, you’ll be waiting for a room to be cleaned and be ready for your use. You could end up waiting all the way into early afternoon.

Thus, the general guidance that if you want the room early, book the additional night.