Interesting idea. But you'll lose the static the moment you touch your faucet.
Further more(if I remember my science classes correctly): the standing up is because all the hairs repel each other(their charge is the same). However your stubble isn't long enough(I hope ;-)) to have this effect. Also, they are too far apart(compared to head hair) for a normal static charge to influence eachother.
The logistics might be difficult. Hanging onto the cat and rubbing him periodically while shaving and fending off attacks. It might make a good comedy movie.