The area with some restaurants more near the TEEM Venue is around ALFAMBRA Street, for example, the restaurant TRITON, about 5-10 minutes from the Venue, and others in the JORDI GIRONA Street (informal sites: Frankfurt Pedralbes, a typical place, as well as others.
Around the area of the el CORTE INGLES of DIAGONAL (Maria Cristina metro stop, or TRAM stop), we can find another area with a lot of restaurants, between SABINO ARANA St. and Gran VIA de Carles III (zone 1), and walking 5-10 minutes from this stop: Gandesa St, Galileo St, Europa St. (zone 2).
Finally, if you take the TRAM from the VENUE to Numancia Stop or L’Illa Stop, you can find a zone with a lot of lunch and dinner options: on the floor (level -1) of the Shopping Center of La Illa Diagonal, as well as in its surroundings.
Taking into account your hotel, and the VENUE, if you are thinking to make almost 7 travels in public transport we recommend buying the CASUAL budget, 10 travels for one person (unipersonal fare) that allow changing between BUS, METRO, and TRAM in the 1h15m of the travel.