We used to drive down overnight to the VGC to avoid traffic from Northern California. Often would leave around 10p to get in around 4a, had to be at the hotel before 5:30a to not get caught in an LA snarl. When we did this, we would check in, grab our room keys so we could use MM, then go crash in the car for the hours until the park opened.
For a 1a arrival, I'd probably opt for a nearby cheap hotel for those hours to get some real rest before trying to go to the parks. Even a few blocks away Garden Grove or hotels by Angel Stadium get much nicer and cheapers than the Good Neighbor hotels near DL.