To run anything on a battery, the item has to work with the same voltage. Your car battery is 12v, so you need a pump or powerhead that runs 12v. Some fishing stores would sell an airpump designed to connect to the car battery terminals to keep bait alive, for example.
If you bought a computer UPS (like the APC 1500) and plugged in a nice airpump, you could create bubbles in your tank for a long time. If you plugged in a Mag pump, it might last 3 or 4 hours. If you plugged in a Tunze, you might get about 8 hours. It just depends on the product, and how much power it will pull to determine how long the battery will hold up.