Can I use a PIR sensor to turn on an arduino?
Yes you can.

It all depends on how you are turning the power on.  You sensor needs to flip a 'switch' rather than actually power the Arduino directly.  (I'm sure you know that but just in case).

So you need a normally open relay in between you power source and you Arduino.  Normally open because you don't want it powered unless you 'turn it on'.  Your relay might be able to be switched by your PIR sensor's data output, but if you need more current then you are going to have to use a transistor and draw more power from the power source.

You have the PIR sensor, so now find the relay capable of carrying the maximum current you need for the Arduino.