Best way to travel from port to BCN airport?

Denny is right, contact cruise line, I seem to remember it was half hour from airport to nearby the harbor where our hotel was using a taxi. I didn't find lines too long at airport but we were off season.

Providing you do not have many delays getting off the boat, you should be fine. While both the Metro and Buses go to the airport, you may want to spring for a cab.

What time the ship docks, and what time passengers are cleared for disembarking can easily be two different times. I'd contact the cruise line and ask. Good luck. A taxi would be the way to go.

Take a taxi. They will be plentiful. At that time of day, traffic isn't an issue. It took us about half an hour to BCN from a hotel located a mile from the port. No hassle with bags either, except at the airport.

We just did the exact same thing from the cruise dock to the airport. We had 7am arrival and 11am flight and made it no problem. Here is how you do it. Make sure you "pack light" which will allow you to "carry off" your luggage from the cruise ship. If you let the ship take your bag the night before, then you will be delayed and might not make it. If you carry it off, you can walk off shortly after 5am. We took the blue bus to the Columbus monument and then subway to Placa de Catalunya where we got on the airport bus and made it to BCN with plenty of time to spare. Having said all that, given your arrival time is 5am, I would double check to see if the bus and subway run that early. If not, you might be safer to just take a taxi. There should be plenty of them lined up when you get off the ship. There is a good chance you could be at the airport by 6am if you take a taxi.