What airport are you leaving from? That makes a difference.
I would use a taxi if you're leaving from SNA or LGB. Just have the front desk call one for you. You should be at either of the above airports about 1 1/2 - 2 hours ahead. If you can, check in online the day before and print up your boarding passes. I always use SNA and leave the hotel 2 hours prior to flight time (that's having already checked in with printed boarding pass). The advantage to the taxi is you don't have to worry about any other parties and you don't have to include the 15-minute waiting window--the desk will let you know when your taxi has arrived.
Taxis, buses and shuttles can all use the carpool lane so it doesn't take that long to get there. It's about a 20 minute trip unless you're in heavy traffic, but I don't think you'll have to worry about that at 4 AM.
When I use SuperShuttle, I reserve it ahead of time online so I can use a discount code and prepay, including tip. The online form asks for your flight time and automatically figures in the pickup time for you, but I usually put down a later time than my actual flight because I don't like getting to the airport with 2+ hours to spare since I'm already checked in.
If you're flying out of LAX, I can't give you much advice as I have no experience. I just know a taxi would be pretty expensive.