This is exactly what I did while in Barcelona for a day. The cruise docked close to city center, making this port perfect for seeing the city on my own.
TIP: If I had to do it over again, I would get off the ship a lot earlier, take a taxi (although this can be expensive) to the cathedral and get a skip the line tour. Then catch the Hop On Hop Off bus from here to see the other sites. The lines to get in to the venue start early and are quite long, so if you don't get there early enough the wait times will eat up too much of your valuable shore time.
The park was originally built as a residential development, but the concept failed and was later turned into a park. Also a UNESCO World Heritage site, the park displays beautiful works of mosaic tile. The dragon fountain welcomes you at the park entrance. Interesting shaped walkways take you to the top of the park and a lovely terrace with mosaic tiled seats. The terrace also offers a wonderful view of the city.
The Gothic Quarter offers plenty to see and do. If you have been to Barcelona before or would rather not get on the tour bus, this area is a perfect place to spend a half day or more exploring.
Barcelona is a perfect city to explore on your own. If you enjoy walking tours, another option is to hire a local guide for a few hours for an insider's view of the area. There is plenty to see and do close to the cruise port, so enjoy what the city has to offer!