Did Alice cooper eat a bat head?

No. Alice Cooper has never eaten a bat head.

However, Ozzy Osbourne (who is 1 year younger than Alice), has "eaten" a bat head.

It happened like this:

Ozzy was a bit high on drugs and was doing a concert somewhere when some fans brought in a live bat to the concert. They threw the bat onstage and Ozzy mistook it for a fake rubber bat. So he figured he might as well bite the head off. Well, that is just what he did. Needless to say, he spit out the head when he tasted blood.