She doesn't sound like the kind of person who will be receptive to you pulling her aside. Doubling up on Tylenol dosing is a med error -- if you documented the dose you gave, she made a med error by administering another dose. This gives you the opportunity to go speak with the unit manager, which will be a perfect entry into your concerns about her attitude and toxic behaviors. What do others on the unit think of this person? Despite her being on that floor for many years, it could be that she's the "trouble-maker" of the unit and needs direction from management about improving her behavior.
I would also recommend reading the book "Communicate with Confidence" by Dianna Booher; in particular chapter 12 "Taking the sting out of negative feedback or criticism leveled at you" and chapter 18 "Responding to insults, boasting, insensitivity, gossip, and other goofs hurled at you". This may give you some insight on things you can say or do to put her in her place. It's not okay to criticize a staff member in front of others, as she is doing to you. If she has constructive feedback, she should be pulling you aside.
I definitely recommend speaking with the unit manager as soon as possible. And when your month is up, I'd get the heck outta there!
Sorry you're having such a difficult time with this person. You'll come across people like this all the time throughout life though, so it's good to learn how to deal with them. Again, I definitely recommend the book above. Other good books for personal and professional growth are as follows:
"Stop Workplace Drama" by Marlene Chism
"How to Win Friends & Influence People" by Dale Carnegie
"Crucial Conversations" by Kerry Patterson (and multiple other authors)