Here's another thing which can help.
Have a plan for worst case scenario. Don't blame yourself, the other person, or anyone. Just try to plan it out so that you handle the situation as an adult and you do not get overly emotional. So, if you ask her and she just says "no" politely, then be polite back and say thank you for being clear with me. I hope we can be friends, and then just walk away and keep communication minimal for a week unless she initiates. After that gap, then just approach her only as a colleague (?) or platonically.
Now that I think about it, generally speaking, it's not a good idea to date co-workers period unless one of you is quitting the job or you don't work with each other directly. (If you are married first, then you end up in the same company, then that can be okay given your responsibilities don't fall along with your partner's, or they would not be considered in appropriate given the circumstances and context of the workplace.)
If the person you ask out treats you the same way this other person did when you were younger, you need to be prepared to stand up for yourself. Be prepared to stay calm. Try to train your mind to do this at least at that moment, and then when you're alone in the privacy of your own home or among close, trusted friends, cry your eyes out.
So, say the person was to laugh at you in front of everyone. Then, be prepared to say something like if you don't to go out with me, that's okay. But if you're going to be intentionally demeaning about it, I have no interest in you either. You're a nasty person and you can go to hell or something like that.
I get the impression that the person who was mean to you was when you were both kids, and now you and this other person are older. I might not have the context 100% because I'm trying to get back to work, lol, but that's the feel I get. Being kids (kids can be cruel) and being older, but possibly more independent and responsible, can definitely make a difference in interactions and possibilities good and bad.
Hope that helps.