I am so sorry. First days of work are always hard. And this is obviously so much harder. Just remember, your response to her being mean is very normal. I know that doens't help, but it doesn't scream "Mental health issue". It may feel like it does. What it says is, that girl is really mean and the girl she was mean to had a normal (noting normal) response to being "abused" (using the term loosely) by a coworker.
Word may spread if something happens. But the focus wouldn't be on you. More like "Did you hear what X did? She was horrible to the new supervisor. She did A,B, and C. And she really upset her." I know thats not ideal. But im not good at giving false, fluffy answers.
We are learning a lot about this stuff in social psychology (Im a senior pysch. major). I know it doesnt feel like this. But if something happens, its not on you. Its on her. And everyone is going to know that.
You cant do anything wrong here. Everyone knows she is mean. If you get upset, that's ok. If you don't, that's ok. You are in charge in that store, and you are the "nice one". (Think Star Wars jedi, nothing they can do is wrong - they're the good guys).

Good luck. And tell us how it goes.