The question of Oedipus the King as a mystery depends entirely on its audience. In the form that Sophocles framed it, the play would not be a mystery. This is due to the fact that it was a common story in Greek mythology and everyone in the audience was aware of the final plot twist before they even entered the amphitheater. This same interpretation of the play also applies to those who, before they read the story, have the ending revealed to them. Because the story is so highly alluded to in our culture, this is not an uncommon occurrence for readers of the play. That being said, I suppose, that for those who had never encountered the myth, long before Sophocles’ time, when it was just beginning to be circulated from Greek city-state to Greek city-state, and also those whose first reading of it is completely free of plot spoilers it could be considered a mystery. However, it should be noted, that the play would likely be less effective in this form, seeing as Sophocles played with the common story and focused on the secret that would be exposed and manipulation of dramatic irony to affect those who would view it as opposed to a story built around the discovery of a mystery.
Despite the entire plot of the piece revolving around solving the murder-case of the deceased King, the real horror doesn’t come from the murder or discovering that our noble protagonist was, in fact, the murderer. The shocking aspect is introduced because these acts were in fulfillment of the prophecy with disgusting repercussions. So while, it wasn’t truly intended to be classified as such, if the reader or viewer is unaware of the horrible reality of Oedipus’ situation until it is revealed in the play, then it would be considered a mystery (though an easy one to solve.)