It's usually "on short notice". In some situations they are the same, but "on short notice" means that you are notifying someone (giving notice) and usually asking or requiring something of them. For example: "I need a babysitter on short notice" means that I have to find a babysitter and I won't be able to tell them far in advance. "At the last minute" means just before something is about to happen, and maybe it doesn't involve someone else. For example, maybe you are leaving on a trip and decide, just before you walk out the door, that you want to wear your boots instead of your running shoes. You could say "I decided to change my shoes at the last minute."