I don't know if this is feasible, but I have a very small 40mm fan which is rated at 12 V/0.07 A, and the Arduino really can't run it with 5 volts.
You need to look at voltage and current. The fan needs 70mA at 12V.
You'll need to use a transistor to connect it to an Arduino. Some how I suspect others have done this. (https://www.google.com/search?q=control+12v+fan+with+arduino
I plugged it into a 9 volt battery and the power worked great.
FYI. Those small 9V batteries are lousy at running motors (like fans). It'll only last 4 or 5 hours if the fan draws 50mA (de-rating since you aren't running at 12V). You'll want to use a AC to DC 9V supply.