There is now a new terminal in Barcelona, the T1, which is slightly further away from the old airport.
The airport is about 13km away from the center of Barcelona, and there are trains, a special...
It is between 15 and 18 km between the airport and the hotel depending on which way you go and by car it will take between 20-30 mins. A taxi will cost about 30 euros or alternatively there is a bus...
Just about 20 minutes by car. Using public transpotion, maybe 1 hour, as you need to go from airport to the center of Barcelona and then towards the port. .