12v fan after power cut
A capacitor or a battery will both work. The idea is to maintain a device that stores electrical power to be delivered after a loss of the main power.

My first attempt would be to use a rechargeable battery. An appropriately sized 12V battery, or combination of lower voltage batteries connected in series, will be able to power the fan. Your fan only requires 0.011Ah to run for the 3 minutes stated. Most batteries should be kept above 50% discharge. So plan for 0.022Ah or larger.