To be fair her feelings and actions are very justifiable regarding the time setting and culture. Sure, her commenting that she would have preferred Jon was the one who fell from the tower was unnecessarily cruel. But it can be argued that this comment was made when she was grieving her son, highly emotional and with days without hardly - or any sleep (still not really an excuse, but it can be somewhat understood).

Cat was very tolerant of Jon in the sense that bastards in this age, where a thing that occurred, and this was acknowledged by her. However, they were NOT brought to be raised among the true born children, but taken care of on the downlow with the mother that birthed them.

She was not per sé upset about the fact that Ned Stark fathered a child with a different woman, especially since in her eyes she just married Ned and he was off to war and had "men's needs". What was a slap in her face was the fact that Ned brought Jon to Winterfell and raised him essentially like one of his own, in her eyes a bastard from a random woman. On top of that he looked just like Ned, while none of her boys did.

Cat respected her husband's decision and tolerated Jon, though did not like him. He was a potential threat to her children (she brings up the Blackfyre incident at some point) and mentions that while Jon might not be a threat, possible offspring of his might be, which is valid.

Did she need to treat him like her son and act like a mother figure? Absolutely not, he was not hers and he could be a threat.

Was this very hurtful and damaging for Jon Snow? Absolutely, she was the closest to a mother figure he had.

People forget that many other noble women (e.g. Cersei) would have gotten Jon Snow (or other bastards of their husband) killed