Yes, 0pen 24 hours with a shuttle bus...You'll make it!
Shuttles at RDU start at 4:30 AM when the facility opens. It's a 5 minute trip to Southwest flights, about 8 minutes to terminal 2 where everyone else flies from. Generally the shuttle never run more than 15 minutes apart, at peak periods they're not more than 5 minutes apart. You might be cutting things a little close if you don't have TSA precheck and are checking a bag. I suggest you call the local office and confirm that you'll need a shuttle ride at exactly 4:30 AM.
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