Yes you can fly into any airport you wish - you'll just have higher transportation costs to the port if you go into FL for a Miami cruise.
We use Southwest that only flies into FL so we have to take a cab over to Miami - it ranges anywhere from $65-85 depending on the cab driver - ask the price up front!
We use cabs because we don't like waiting around for the shuttles to fill up. (the only leave once full)
No idea on the private shuttle pricing, it's only been the 2 of us so a cab has been fine.