Hi,
Normally I would never tell anyone I work with that I have a drinking problem. There are too many prejudices against alcoholics, by people who don't understand - namely, Why don't you just stop drinking?
In this situation, I don't know what I would do. Coming clean might help your case for keeping your job, but it might backfire. Since honesty is always a good idea, I'd probably come clean and try to impress your boss with the fact that you are getting help for your addiction.