LAX Airports are not a good base, and I would definitely use Uber/Lyft to get their for Early Entry, don't forget you need about 30 minutes before opening to deal with the security check and getting in line.
You could use Starline Tours, but you won't get to Universal until the late morning.
As for true public transit, Hotel Shuttle to LAX, take the free G shuttle to the Green Line Metro Rail station, take the Green Line to the Blue Line and then the Red Line to Universal City, walk across the street and take the free shuttle up the hill to the Security Check area. Way too many transfers and time.