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The flight will in all probability be arranged by Celebrity but if you want to be absolutely certain ask your agent. If that is the case most passengers fly out the same day for Mediterranean cruises and the ship will wait should there be any hold up. You can arrange to have extra days in Rome prior to your cruise and Celebrity will arrange hotel and transfers or you can arrange it yourself in which case the onus is on you to get to the accommodation and then the dock. Don't worry though thousands are doing exactly what you are doing all the time..... Nei.
It depends. If the TA has booked the whole package through Celebrity you will fly out the same day, and as the fligths are with Celebrity you are covered if they are delayed etc. The easiest way is to look at your booking confirmation from Celebrity to see if the fligths are shown. If the TA has booked the flts independently and the flts are late then you are on your own.