Unfortunately, this is a very difficult question to answer given the disputed understanding of what precisely Aristotle means by katharsis. In the Poetics, Aristotle writes:
Tragedy, then, is an imitation of an action that is serious, complete, and of a certain magnitude; in language embellished with each kind of artistic ornament...; in the form of action, not of narrative; through pity and fear effecting the proper purgation [katharsis] of these emotions. (S.H. Butcher translation)
Butcher takes katharsis as meaning "purgation", but such an interpretation has been hotly disputed by modern scholars. If we were to accept Butcher's translation as the correct one, though, and apply it to Sophocles' Oedipus the King, would the playwright's audience have come away from the theater that day with the emotions of pity and fear properly purged?
Maybe it's just me, but I can't help feeling pity for Oedipus. After all, he has just discovered that he is guilty of some of the most horrific actions imaginable. Furthermore, the audience knows that Oedipus is going to be exiled from his native land (which was a much "bigger deal" for the ancient Greeks than it would be for a modern American). Also, Oedipus' access to his children, especially Antigone and Ismene, is going to be regulated or perhaps cut off completely by Creon. Thus, it's difficult to imagine an audience member having their "pity" purged.
As for the emotion of fear, I can easily imagine a person walking away from the theater being very fearful. After all, Oedipus tried as hard as he could to avoid what Apollo said would come to pass. Nevertheless, Oedipus still killed his father and married his mother. Again, it's difficult to imagine someone going away from Sophocles' play with their emotion of "fear" purged. It seems to me that a person would leave that theater even more fearful or reverent toward the gods.