I take it that you are booking cruise only and making your own flight arrangements?
If so, I would strongly suggest that you book a flight to arrive the day before and check into a hotel overnight. This is to build in a buffer -If your flight is delayed for any reason you really dont want to miss the ship. Boarding the ship will usually commence around 11am/noon. Disembarkation will take place very early in the morning and everyone is usually off by 8.30/9am latest.
I would never fly into a US airport from the UK on the day a cruise departs. Flights from the UK tend to land in the afternoon - if it is a few hours late you miss the boat! However if you are doing a domestic US flight then I would advise landing asap. The port is at San Pedro (pronounced Peee droh) which is about 40min drive from LAX and you can either purchase RCIs costly transfers or arrange your own with a company like Supershuttle dot com or primetime shuttle. When docking everyone will be off by 8.30am latest so tell them it is an early pick up. I suggest that you join the meet and mingle for this cruise and you can make friends before you sail, and pick up tips for what to do in each port.
As previously suggested if your inbound from the UK I would err on the side of safety & fly 24 hrs earlier. If it's an internal domestic flight then there are normally a few more flight options should things not go to plan. I did a 'same day' flight into LA from Las Vegas, only about a 30 minute flight. I tied my Cruise labels on to the luggage in the airport & was immediately accosted by the cruise 'reps' & was given the option of going with them, so I checked out the taxi price & it was exactly half the cruise transfer cost. One point missed by the previous comments, yes you will be off by 8.30am but allow for the dreadful immigration procedures which can add another hour to your disembarkation.