Explanation:
Remember, an ironic statement is one in which what is been said is to be understood in another way. In other words, the meaning of a statement is actually the opposite of its literal meaning.
So what Scout actually meant by her statement is that Aunt Alexandra trying to bring her into prison by wanting to turn her into a lady.
Hence, the expression "starched walls of a pink cotton penitentiary" refers not to a physical penitentiary but to the perceived prison-like restrictions of dressing like a lady.