Do you like art museums? If so then my advice is to take a taxi to the Museum of Islamic Art on the Corniche. It's a world class museum and the largest collection of Islamic art in the world. It's maybe 10 minutes from the airport. A couple hours would be enough if you moved through the exhibits at a decent clip. Hand over the batteries and adapter to the attendant at the museum and I'm sure they would be happy to charge them. Museum is free and they usually have traveling exhibits loaned out from places like the Louvre and the Rijksmuseum, which you can also see for a fee. The museum architecture is spectacular.

http://www.mia.org.qa/