Since you are in Terminal 2B it is handy to take the train to go to the city center.
The train stop is 10 minutes walking from the terminal. You just have to walk through a long-elevated corridor (it crosses a big road).
You can find the timetables in Rodalies Renfe's webpage. Select Aeroport as Departure Point and Barcelona-Passeig de Gràcia as Destination. For this you can use your T-10 ticket to pay for the trip, contrary to the metro that needs an extra payment.
The train works every 30 minutes and the trip is just 26 minutes. When you arrive, you are 5 minutes far from the central Plaça de Catalunya and 15 minutes walk from Liceu, through the Ramblas. So it is a nice, turistical walk to your destination.