Just want to make sure you know that LA County USC Medical Center is not part of USC's campus which is just southwest of Downtown LA. County USC is southeast, near East Los Angeles, a few miles away.

The LAX Flyaway bus from LAX to Union Station is a pretty good option. Union Station to County USC would be about a 15 minute drive, even with the possible rush hour traffic (and driving from East LA to LAX on a Friday afternoon is more often then not a miserable experience).