With one day, personally I would not waste time on the Picasso Museum (unless you're an absolute Picasso fan). But it's worth taking a walk in that general area (El Born).
What I would add is La Pedrera on Passeig de Gracia, with a visit to the roof.

And while there might be some places for a nice dinner NEAR (not on) Las Ramblas, make sure that you know where you're going, to avoid the tourist traps.