Easiest, quickest, simple way it to take an uber, lyft or cab.
And best part - you DO NOT HAVE TO MEET THEIR SCHEDULE.
for example, if you schedule a 9:30 am private transfer, you are FORCED to adjust your schedule on the last day to insure you not only get off in time, but clear customs on time, etc.
When booked via the cruiseline, they then book it with another party. Way too many places for a screwup in my mind.
It is only 15 minutes to the airport. Easy stuff.