1 - Sagrada Familia, I would recommend to go inside but skip going to the top. You can walk or take the elevator but you are press for time. Give yourself 1/2 to study the outside statues and both sides of the cathedral. Make sure Passion Façade side of the cathedral. The door the entrance is fantastic.
2 – From Sagrada to the train to Casa Mila. You will have to wait on line to pay to enter, hopefully it will be a short one. Go straight to the roof see the beautiful chimneys then walk down see the Modernism furniture inside the apartments.
3 – Walk from Casa Mila to at Plaça de Catalunya, should take 20 – 25 minutes and it will give you an idea what the Eixample district is like. Also you will pass the Island of Discord and will see some more Modernism buildings.
3 – From Plaça de Catalunya, walk down the Ramblus in the afternoon and check out the street performers.
4 – Half way of the Ramblus stop at the La Boqueria market. The market is very colorful and energetic place.
5 - From market walk over the Barcelona’s cathedral which it 10-15 mintues away and it it give you an idea what the old city is like.
Walk fast like a New Yorker and you will do ok!
Bill