First question: do you have four hours to kill, or four hours arrival-to-departure? If that's the case, and you're arriving on an international flight, you'll really only have about two hours to kill.

There is absolutely nothing to do in the airport aside from the lounges or getting a really good coffee. Unfortunately, there is nothing to do nearby either. The city is about 35 minutes by train or taxi.

Provided you have the time, I'd suggest grabbing a cab to the center, get out and do a little wandering around. Have it drop you at Plaça Catalunya and you can walk the Ramblas to the sea in about an hour, more if you take your time and stop for tapas a time or three. From there, you can catch another cab back to the airport. It will run you around 20-25€ each way. The train is cheaper, and you can take it to the Pl Catalunya station then take a cab back if you'd rather.

There is lots to see along the Ramblas, though it is very touristy (and very pickpockety, so be careful). Given the time you have, it's probably your best bet. Other options would be a quick tour of Montjuic by taxi, though that will be expensive.