You might call them and ask what time their computers go down for backup. I had that experience at BW once when we had a flight delay and needed to check in during the wee hours; we had to sit in the lobby for an hour because they could not access the system, so they could not key a room for us.
Generally, when you show up in the wee hours, you don't get premium rooms; you get whatever is still vacant. However, if the hotel is really full and your reservation is guaranteed for late arrival, you *could* get upgraded, though that is usually a real long shot. It did happen to us once in at the Chicago Hyatt; we checked in at 2 am on a Priceline ressie, and they gave us concierge because it was the only thing that they had open.