Hi Pal,
The Port is listed as Puerto de Barcelona. If you are going from the international airport, you will have no trouble getting a taxi just outside the baggage terminal. The fare, a month ago, was about 13 euros plus add on for baggage and tip was less than 18 euros. Depending on the location of your hotel, taxi fare would be from about 7 to 12 euros. The port is in fact in the city, and you can see the terminal from the foot of the Ramblas. If you are going to the cruise port from the airport you might want to buy some euros before you leave home, or check out the location of the ATM in the airport. It's handy to have some one euro coins for tips handy.
Peter