Dowdeswell! We are also on the same cruise as you! Good question, as I was wondering the exact same thing, so glad I came across this site and your question! My sister left for the same cruise we are going on yesterday, and there flight was cancelled due to bad weather in new york, they had to fly to amsterdam, then to new york when the weather had cleared. So they only got to NY late last night. Hopefully weather will have improved by the 31st!
Once clear of customs the Royal Caribbean reps will be there and take you to your transportation with your luggage either going in the coach you are in or on a van and delivered to your hotel if you are staying overnight before joining the ship the following day. Make sure your ship ID slips are attached to your cases before you part with them at Newark. Some passengers like to put them on before they leave home in case they get lost, but so far we have never done this.
We are planning to go in to New York, but do we have to check in the hotel first with our luggage or can we go direct to NY while our luggage goes to the hotel? The reason I ask is because we land at 12.00 midday but cannot check in until 15.00.
By the time you land and collect your cases and get through immigration etc meet up with the rep then you will have to go to the hotel to check-in and by that time 15.00 may well have past by, thereafter you can go out and enjoy the time you have. You will receive details from the rep or at the hotel about the transfer to the port unless you have made your own arrangements. We were last in NY two days before 9/11 and we were actually at the buidling itself but did not have time to go in. We were a bit disappointed in that things looked much smaller than we thought even the Statue of Liberty wasn't as tall, but we still enjoyed our visit.