The thing about the Picasso Museum is that it can be incredibly, unbelievably, crowded. It was like that in 2016 and I can only assume it has not improved. It is possible to enjoy architecture in crowded conditions (e.g., Casa Batllo or Casa Mila) because much of what you're looking at is above people's heads. That is not true at the Picasso Museum. I was really frustrated at having to push through a mob of people to get close enough to read the labels posted beside the artworks. It's not that there was a crowd of people in front of one or two iconic paintings (that's common if a museum has a few well known works); the entire museum was packed with people except for the room containing the ceramics. So unless you're a big fan of Picasso, I urge you to skip that museum and go somewhere that is easier to enjoy.
I, too, am a fan of the MNAC. In addition to the paintings and sculptures it has magnificent frescoes rescued from churches in the Pyrenees and an interesting modernism collection that focuses on decorative arts, furniture and jewelry.
I agree with Carlos's other recommendations as well, though Parc Guell and Sant Pau are quite different experiences, and I didn't think the (admittedly crowded) conditions at Parc Guell degraded the experience terribly. Face it: Barcelona gets somewhere between 20 million and 32 million visitors a year (I've seen different numbers), so you will be among thousands of fellow visitors in Barcelona. At least don't waste any unnecessary time in line. You'll still probably have to queue up briefly even with a pre-purchased ticket.
Note that you'll need to pre-purchase tickets for any of the sites listed below that make the final cut because ticket lines can be more than an hour long, and sell-outs are distinctly possible. I recommend making one of the problem sites your first stop each morning, because the tickets will all be timed, and it is not so easy to figure out how long you'll want to spend at Site A or how long it will take to travel from there to Site B. At least you won't have such problems at the first site each day.
La Sagrada Familia
Parc Guell (tickets not sold on site at all)
Palau de la Musica Catalana (English tours may sell out)
Because your visit is so short, you really cannot wait until you arrive to try to score tickets. The scattered tickets that might be available would perhaps not fit into a coherent sightseeing plan (overlapping times, etc.) First-time-slot tickets are usually especially popular because--as noted above--they somewhat simplify a tourist's schedule.