Can I use 9V battery to power up a cooler fan?
If your cooler consume less than 1A (it's writen on it) and if you don't care so much about power consumption, you can use two 9V batteries and LM7812 - it reduces output voltage from 18V to 12V and have only three pins: In (+), Gnd (-), Out (+)
https://www.fairchildsemi.com/datasheets/LM/LM7805.pdf