Running 12V PC Fan with 9V Battery
Yes, you can do it, but the battery won't last very long. 

It will probably run at about 3/4 the speed and draw about 3/4 the current (rough estimate) so around 100mA. 

Looking at a typical datasheet for an alkaline 9V battery, we can see the capacity will be perhaps 350mAh (down to 4.8V) so it might work for 3-5 hours, as the current will drop as the battery discharges. 

If you were to step up the battery voltage so it was 9V continously, the operating time would probably be less than 3 hours.