I think that's cutting it WAY too close. For a 4:30 set sail, I believe you have to be checked in by 3:00 pm. That only leaves you 2 hours to get bags, get to the port, get through security at the port, and get checked in. Check with Royal Caribbean, and see what they recommend as the latest arrival time for your cruise. You can find that info here
, and just select your ship, sail date, and embarkation port and debarkation port from the dropdown list. The embarkation port is where you are boarding the ship, and the debarkation port is where you will get off the ship at the end of the cruise.
Also, bear in mind, that since you did not book air through the cruise line, they will not wait for you if for whatever reason you're late, and you will have to catch up to the ship at the next port of call at your expense.