The only way that I can tell you to be prepared is to make Paperwork, and that will only prepare you for what you know is coming.
I'll be quite honest, I've never come into a situation that I was too panicked to handle. It's my personality.
There's different levels of "emergency"
First there's the superfluous emergency. Which is forgetting a prop hand-off, calling a cue late/early, missing a quick change. It sucks and it affects the pace of the show, but it's easy to fix. If it's the first time, that's ok, make a note of it, and do it right the next time. This mistake, I don't think it needs to be told to everyone that it happens, if you can fix it. If you keep messing up either a)reconsider your career choice, or b) buck up, do the job you're getting paid to do. If you constantly make mistakes, on the same thing, fellow SMs won't trust you, actors won't trust you. A friend of mine was not cut out for stage management, but wanted to pursue it. She kept forgetting a handoff for an actor, this actor lectured her after the show, went to the SM and things were shuffled around, so someone else was put in charge of that prop. It sucks, but you're getting paid, and entrusted to do a job, you need to do it.
There are then, non-emergency emergencies. They are something that happens, that need to be fixed and can seriously affect a show, but non life threatening. If something out of the ordinary happens, I say"...ok..." stop for 2-3 seconds, take a deep breath, and as I'm briskly walking to fix the situation, I am on my headset letting everyone know what's going on. This can be a piece of furniture broke, mic pack not working, costume piece/prop broken or missing. or "small" medical emergencies. I had an actor who decided to help Stage Management and the Crew by moving his instrument by himself, which is a big double bass, and includes stepping down 16 inches with no intermediate step. He stepped oddly and the bass hit his head, and he got a cut. It was near the end of a tech rehearsal, so he wasn't needed, the other ASM stayed with him making sure he was fine and didn't have a major concussion. The PA ended up taking him to the hospital, where he got stitches, but did not have a concussion. I consider this a small medical emergency because in the end, he was able to perform, and he was only out for a few hours. Also, it was handled really well.
And then there are legit emergencies where you don't have time to second guess. They are life threatening, and if dealt with wrong. You have everyright to panic, but calmly. You need to deal with the situation, and then you can have a venting session, preferably with a drink, afterward.
I had an interesting summer where I learnt a lot about dealing with these situation, and equity rulebooks.