Port Vell puts you close on paper. Which terminal you actually land at still decides your morning.
Barcelona is one of the few Mediterranean ports where the ship can genuinely dock close to the city centre — but "close" depends entirely on which of the port’s several terminals you land at. Some sit a fifteen-minute walk from Las Ramblas; others are a much longer walk around the World Trade Center complex, with a free shuttle bus doing the work instead.
The shuttle drops most passengers near the Columbus Monument, at the foot of Las Ramblas, which puts the Gothic Quarter, the waterfront and the old city within easy walking range. A taxi or the L3 metro line covers everything further out — the Sagrada Família and Park Güell in particular are not walkable from the port.
The Sagrada Família and Park Güell both require timed tickets that regularly sell out days ahead, so book before you sail if either is the priority. If your time in port is under five hours, the Gothic Quarter, the Boqueria market and a slow lunch cover more honest ground than trying to add a third across-town sight.
Barcelona is also a common embarkation and disembarkation port for Mediterranean sailings, not just a call on the itinerary — our own Mediterranean pick uses it as one of its round-trip ports. If you are starting or ending a cruise here, an extra night in the city either side is a better use of the airfare than a single rushed afternoon.