I must admit to being a subscriber to the traditional view that the acropolis of Antioch was well within the city, in most likelihood just above the theatre on a spur that ran out from the lower slopes of the mountain. My intrepid colleague, Jorgen Christensen-Ernst, likes to challenge the traditional though and he is on the spot in Antakya all the time. Their have been no excavations at the acropolis (as with virtually everywhere else in the city) so no-one can say definitively either way.
Jorgen did however recently stumble upon some references that he believes might signal that the acropolis was much much higher up and that, in fact, it occupied the place where the fortress now sits perched upon the mountain. An argument against such a siting is that the acropolis would have been well-nigh inaccessible to anyone in the population except those with the fortitude of a mountain goat. Certainly before the major walls were built (which was NOT under the Seleucids) that would have put the acropolis outside the city fortifications. A caveat that offers a pro case for the fortress as the site is the city of Priene where the acropolis is so vertiginously elevated that it is pretty much out of sight of the city way down below.
In defence of the fortress option, Jorgen found two references in the book Asianic Elements in Greek Civilisation: The Gifford Lectures, 1915–1916, Edinburgh by William Mitchell Ramsay in which the author states: "That there was a Katoikia called Koloe in the neighbourhood is certain from the inscription on the accompanying monument, which is in many respects the most important of all. Despite the resemblance of the ancient Koloe to the modern Koula, the late Byzantine evidence shows that Koula was understood as the Turkish, and probably old Anatolian, kula, kale, a fort or castle.
There is a distinction now made between koula, tower, and kala or kale, strong place, fortress. The term koula, kula, is explained by Ducange, Notae in Alexiadem, p. 621, as applied by the Greeks to all acropoleis. The acropolis of Antioch on the Orontes was called Koula by Anna Comnena, ii. pp. 89 f, and Kala is mentioned as a strong tower on the west side, by Scylitzes in Niceph. Phocam, quoted by Ducange, loc. cit., which shows that the words are practically identical. In all probability the words are variants of an old Anatolian word, taken over by the Turks; but H. Kiepert in a letter to me preferred to consider them early Turkish words".
We would note that Anna Comnena lived from 1083 to 1153. In her childhood Antioch was still under Arab rule and then passed to Crusader rule. Its hard to believe that she actually ever went to Antioch, so her observations are most likely second-hand.