My flight does not arrive until 11:45 pm on a thursday night to RDU, can I still get an Enterprise car rental and what about shuttle service from RDU to rental place that late at night?

If you call number provided for that location in confirmation/reminder email or in app reservation info they'll tell you hours. I recall midnight to 5 they have alternative pick up process but don't remember details.

Call and ask

They close at 1am so you be able to get a car. I'm not sure what time their shuttle service stops operating.

Enterprise is open till 3 or 4am

I'm not sure about late night service. I know there is a shuttle as long as the rental office is open.