Well, this is one of those “yes, but …” things.
(1) The Archbishop of Milan is ex officio Capo Rito of the Ambrosian Rite; historically, most (though not all) have been “native sons” of the Ambrosian Rite. Paul VI was one of the “not all” in the foregoing sentence: he was Roman Rite from Piedmont and the only reason he was involved at all with the Ambrosian Rite is that he was named Archbishop of Milan. The same is actually true of his great predecessor in that office, Bl Ildefenso Schuster, who was Roman Rite (and a Benedictine) from Rome. It’s interesting to note that despite their non-Ambrosian background, both took the position of Capo Rito very seriously. For example, one of the few positive things I will credit Paul VI for is his post-conciliar personal intervention to halt his successor’s attempt to totally suppress the Ambrosian Rite.
(2) Keep in mind that the Ambrosian Rite is one of the few remaining “territorial Rites” and so its use outside the ancient ecclesiastical Province of Milan is very limited. Remember, too, that it is a Rite of the Latin Church and not a Church sui juris on its own.
(3) While it is within the purview of a Roman Pontiff to use any Rite he pleases, I don’t recall, (and can find no reference to), any public celebration by Paul VI in the Ambrosian. If he physically visited Milan during his pontificate, (and I don’t know if he did), he would likely have used it then. He may, of course, also have used it privately on occasion.