If you're talking about Ft. Lauderdale airport (FLL) it's about a $10 cab ride and maybe ten minutes on a bad day. I believe most cruise lines run free shuttles to the port on cruise days--they certainly used to. If you happen to arrive a day or two early and rent a car, the vans from the major airport car rental return lots will happily take you to the port instead of the airport if you ask them to.
In short, it's extremely close by, nothing to worry about.
Yes, its very close, their properties actually are side by side. When you are flying in, be sure to look for the cruiseships, hard to miss. Agree with AR, a cab would be the easiest/fastest way to get to the port.