I have used the Barcelona port three times and whilst it's do'able it's a long way with luggage. When you depart the DCL terminal, which is Terminal C you have a bus stop outside, where a blue bus "cruisers" picks up every 30 mins which will take you to the Colombus monument which is a stones throw from Drassanes metro station. http://http://www.barcelona-tourist-guide.com/en/transport/barcelona-cruise-port.html
This is a great site for info.
there is not a metro into town from BCN. There is a train that will get to Sants Station that has several metro lines.
The "aerobus" runs from the airport to Placa de Catalyuna at the "top" of las rambles. there is a metro station there that will take you to the Drassanes stop near the Columbus monument. IF you don't have to much luggage you can always walk downhill through Las Rambles to Columbus. We did that from out hotel on our first trip to Barcelona.
If I recall correctly the Barcelona port authority or whatever they are called do not allow anyone to walk into the port area even if they booked on that cruise. You have to take some sort of transportation (bus, taxi, private car, etc).
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