I was the unlucky person who was elected to drive in Barcelona and over to southern France. Getting out of Barcelona was very tricky. I am used to roundabouts but these were quite difficult to manage. There was a lot of road work going on and detour signs were not helpful.
I am not sure about parking as we did not park other than when we returned the car to the train station, but others may know if it is difficult or costly to park. Since you just have one day, you might be better off on the HOHO bus, if it runs on Sunday. No worries about parking and it takes you pretty much anywhere you'd want to go.
I didn't drive to the airport, so can't comment on that. We just had our hotel call a taxi for us. Arranged the night before.

I haven't been to Barcelona yet but I do rent cars on all of my vacations. Last rental drop was in Madrid by the Atocha train station.

What time is your morning flight? How is traffic from the city to the airport on a Monday morning? Will you gas up the car on the way to the airport? Is the return lot close to your terminal or the airport? Are there any savings between having the car for the extra time and parking costs vs. dropping on Sunday and taking a taxi to the airport? Is there convenient parking at your hotel?

I usually drive to my last hotel, check in and drop the luggage then return the car in the city, taking a cab the next morning. However, if my last night is not close to the airport, I drive to the airport in the morning and drop the car.

Since you are not familiar with the city, I would drop the car off at the airport and take a taxi to your hotel. Barcelona can be hectic most of the time, there are a lot of one-way streets, traffic moves quickly, and street parking can be a problem. Having a car in the city is not highly recommended. You can do the loop around the city, drop the car off and should be at your hotel in 30 minutes or less.

Sorry, should have added that while others seem to have issues driving in cities, I'm OK with it and I travel solo so don't have someone navigating me, just my gps.

My biggest issue is with underground parking garages with the very narrow round-and-round ramps that are two-way... seems I'm the one who has to back up to let the other car go by! And, those tiny, narrow parking spaces!!!

Whatever your decision, have a great trip!

DD drove in BCN..I DID NOT!!! She wouldn't do it again. Parking was in pay for parking places (2 of them) and were 30E each night, 4 in total with 2 nights out of town. Turned the car in ASAP and got on our cruise ship. First she and I picked up the car @ airport, drove to Italy and back..Picked up DH at BCN and styed 2 nights in BCN drove to Cardona.Returned to BCN . Should have taken the train to Sitgres and Monserrat but no, wasn't thinking. Bad on us.. Never again. The cruise was great however and so were the places but the car in the city was awful!! the parking places were terrible and the traffic unspeakably difficult and DD is a professional driver.. Live and learn!!!

We have been Barcelona many times but have never rented a car while there. In fact I try to avoid renting a car while in most European cities. And when I do a rent a car, as I leave the city, I try to find a car rental facility near the outskirts of a city, so I do not have to navigate the city.

Coming from France, it is easy to drive to the airport- you don't leave the highway. But driving in and out of Barcelona can be a nightmare and finding a parking place is very difficult. I would do as Robert suggests : drop the car at the airport and take the bus or the train into the city.

Barcelona's fine for driving round: no more difficult than any other largeish European city: far more like Milan than Rome. Drivers are a lot less prone to displays of emotional incontinence such as pointless horn blowing than, for example, New York.

Parking's no more difficult to find than, say, London, with restrictions about as well policed. Parking costs are about normal for larger cities: (€15-€35 a day, or around €2.50 an hour, depending on location). There's far less restriction on car access than in, say, Florence or Oxford but you do need to check whether accessing your hotel is restricted

A lot of the perceived difficulty comes from the narrowness of corners in some multi-storeys. There's also the usual problem with GPS in the middle of European cities: the inevitably delayed response time can often make the oral instructions for finding a specific car park simply wrong, unless you've got a passenger reading the GPS map.

Staying at a slightly peripheral hotel with mostly surface parking (like the Rey Juan Carlos) presents practically no problems at all for the poster (though I've forgotten how much it costs). Staying near the historic centre requires the usual navigation skills and the usual problem of hoiking bags from a parking place some distance away.

City council parking guide at http://www.bsmsa.cat/mobilitat/index.php/bsm-car-parks/bsm-car-parks/network-map

Why not drop the car in the city, then take a taxi to the airport in the morning? Typical cost would be 25-30 euros, about the same as overnight parking and a lot less hassle. Also, it's usually more expensive to pick up or drop at airports.

Be sure to check the hours of operation at rental offices; they often are limited and quirky, although you may be able to drop off at any time.

Are you aware that dropping a car across border, for example if you picked it up in France and leave it in Spain, can incur a hefty extra charge? You might consider driving to Perpignan, where you can pick up the new TGV line to Barcelona (depending on when you are traveling). Even though it seems like backtracking, the fast train may make it the same and a ticket would be far less than the usual drop fee.

Ignore if you picked up the car in Spain.