You will probably have to call for a cab at the train station. Taxis are not that common in the Detroit area, and if any are there when you arrive they have probably been pre-ordered by other passengers.

At the airport security is open 24/7, so you can get into the gate area if you are already checked in for your flight. If you need to check in, the ticket counter opens at either 4 or 5am. You could wait in the public area until then and probably not be disturbed.