It's not about damage, it's about solving the puzzle in the way it was intended to be solved, if you're into dealing damage to overcome obstacles you should play a different type of games ;-) 

I can give you a spoiler if you want ... look below.

You first have to wake the cat with the cannon (use the long fuse), then look it in the eye (this was mentioned on some hint note) by using the action button as soon as it looks up. Not sure anymore if you need any additional items for this to work.