In front of the Picasso Museum is the Textile Museum. On such a short trip to the city, however, it's not worth visiting, though its café is a pleasant spot for a refreshment or coffee if you're in need of one.
Continuing along Carrer Montcada, we come to Passeig del Born. To the right, we can see the back of Santa María del Mar church. It is a must to get into Santa Maria del Mar. Ticket price is 5€, 10€ including going up to see the views (worthy to do it but expendable if you don't feel like doing it).
Walking alongside the church, we pass Fossar de les Moreras, a memorial square commemorating the Catalans who died during the siege of Barcelona in 1714, at the end of the War of Spanish Succession.
We then come to Placa de Santa María del Mar, where we can see the impressive rose window in the entrance to the church. Don't leave without going inside, as this is one of the most beautiful churches in the world. Pay special attention to the reflections that the stained glass makes on the walls.
Outside again, we recommend walking up Carrer Argentería (street that runs from Santa María del Mar to Via Laietana) or having a drink or glass of wine (2-5?) in La Vinya del Senyor, the bar in front of the church.