Unfortunately, I don't generally keep rubbing alcohol in the house. The only thing I have close to it is a bottle of brandy that's a bit too expensive to be using in that fashion... It's mostly for decoration so it's never even been opened.

Thankfully, I wasn't stupid enough to go to the bathroom using that hand. After consulting many websites, it seems like a very common mistake people make so I was properly warned... But I'm mostly fine now -- ice for the last six hours seemed to have worked (I suspect it was more the time than the ice that made the difference though).