Like others, I wouldn't hesitate to visit Barcelona on a Sunday. It's a lively city and there's always plenty to do.
However I'm wondering whether you could improve on those awkward flight times by switching airports? Barcelona is served by three airports. As well as the 'proper' Barcelona airport (the wonderfully-named 'El Prat'), you can also use Girona and Reus. (easyJet, among others, uses Barcelona. Ryanair uses Girona & Reus). Both of those airports are just 'an hour and a bit' away from Barcelona on the express coach. Of course there's nothing to stop you, for example, flying into Barcelona with one airline and out of Girona with another. It ought to be possible to maximise your time in Barcelona.
The city's official website might be of interest to you:
http://www.bcn.es/turisme/english/turisme/welc ome.htm
Chris