Cheap hotels in Barcelona don't really exist. The last 2 years we looked at 'alternative' accommodation & decided on a 'Hostal'. These are a cheap way of staying in town centres but you have to be very selective as there are far more bad ones than good. The really cheapies are used by the student population so need I say more. However the ones we have used have been ideal & have satisfied our party of 6 adults on both occasions. This is the one we used last year, very clean, A/C, En Suite, good sized rooms,100 metres from Las Ramblas (the main boulevard) Here's the site www.hostalgrau.com ........Good Luck
I should add, the port is at the bottom of Las Ramblas, previously mentioned.
We did the same cruise last year. The port is about i mile from the town center (las ramblas)There is a crise port bus for 1euro takes you to all the ships. There are many Hotels just off las ramblas, depending on price. NCL Gem is a good cruise to see the MED. Hope you enjoy yourself.
David and Joy.