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The following Barcelona sights normally require advance-purchased (usually timed) tickets to avoid extremely long lines. I do not know how the Articket factors into that. You won't want to spend 60 minutes, or more, in line at each of the top sights.

La Sagrada Familia (decide whether you want to ascend a tower)
Parc Guell
Casa Mila/La Pedrera
Casa Batllo
Picasso Museum (tends to be a madhouse)
Palau de la Musica Catalana (tour required, offered very often in English)
Camp Nou Soccer Experience (which I believe sells out early)

I recommend a walking tour of the Barri Gotic. I liked the one run by the tourist office, but there are commercial options as well.

For a very impressive modernista complex that usually has no entry line, check out the Sant Pau Recinte Modernista site.

Barcelona has a lot of very good museums; check your guidebook to see what might appeal to you.

Note that you'll probably need at least 90 minutes at each of the above sights except for the Palau de la Musica Catalana. Sant Pau likely needs longer; it's is a multi-building complex with attractive grounds.

Barcelona in four days
Plan of attack with pre booking sights

Saturday 26 May (arrive at 2pm)
•Parc de la Ciutadella
•Barceloneta
•Magic Fountains of Montjuic

Sunday 27 May
•Palau de la Musica Catalana (BOOK a first up visit)
•Bari Gotico - Cathedral
•Las Ramblas - Placa Real, MACBA, CCCB

Monday 28 May
•Sagrada Familia (BOOK a first up visit)
•Hospital Sant Pau
•Eixample - Passeig de Gracia, Casa Mila, Casa Batllo

Tuesday 29 May
•Montjuic - Joan Miro Museum, Castello, Olympic Ring, MNAC
•Plaça Espanya - Las Arenas

Wednesday 30 May
•Picasso Museum (BOOK a first up visit)
•El Born - Basilica Santa Maria del Mar
•La Boqueria
•Park Guell (BOOK a late afternoon visit)

NOTE:
Picasso Museum is closed Mon
Joan Miro is closed Mon
Antoni Tapies is closed Mon
MNAC is closed Mon
CCCB is closed Mon
MACBA is closed Tue