I need help finding examples of classical thinking, those of Neoclassicism/Enlightenment, evident in Book 9 of Paradise Lost. In what ways does John Milton present Neoclassical thought in Book 9?

Strictly speaking it isn't correct to use the terms "Neo-Classicism" and "Enlightenment" in relation to Paradise Lost. Neo-Classicism is a term used to describe a creative movement that emerged in the 18th century, long after Milton's death. And while aspects of Enlightenment thought were certainly in the air during Milton's lifetime it would be somewhat misleading to suggest that he was part of that developing current of ideas.
Nevertheless, Milton was undoubtedly a Christian humanist, a Renaissance man steeped in classical learning. In writing Paradise Lost, he consciously set out to create an epic worthy of the ancient Greeks and Romans, especially Homer and Virgil. At the same time, he wanted his work to be a Christian epic, one that dealt with the theme of man's first disobedience, the primal sin of Adam of Eve. Milton regards this theme as much more worthy of epic treatment than the tales of mortals and gods written by his ancient forebears. But as a Renaissance humanist, Milton still wants to connect his work to the world of antiquity, to place it firmly and securely within a cultural tradition he so deeply venerates.
At the start of Book IX, then, he once more invokes the name of Urania, the celestial patroness and muse of Christian inspiration, to help him tell the story of Adam and Eve's disobedience. One example of the influence of classical literary models on Milton comes in the form of an epic simile, an extended simile in which two dissimilar things are compared using "like" or "as." This is from Book IX lines 634-644:

as when a wondering fire, Compact of unctuous vapour, which the night Condenses, and the cold environs round, Kindled through agitation to a flame, Which oft, they say, some evil spirit attends Hovering and blazing with the delusive light, Misleads the amazed night-wanderer from his way To bogs and mires, and oft through pond or pool, There swallowed up and lost, from succour far. So glistened the dire snake.

So in Book IX the form of the classical epic is still being used by Milton. But crucially the substance is wholly Christian. Milton's take on the Fall of man is being expressed in the style of the ancient poets he so much admires. Book IX epitomizes Milton's Christian humanism, in that it illustrates the way in which the inherited traditions of classical learning can give shape and new life to elements of Christian thought.

Neoclassicism refers to Greek and Roman influences in art, literature, and philosophy. Milton's subject for Paradise Lost, of course, is Biblical, but the tone and the structure of the epic poem harkens back to the Iliad, Odyssey, Aeneid, and other epic poems. In Book Nine, Milton repeatedly makes classical references. He says that he will need the inspiration of the Muse that helped Homer, because the story of the Fall of Man is "not less but more heroic than the wrath of stern Achilles." The Book, as indeed the play in general, is full of other classical allusions. There are references to "Ammonian Jove," "Scipio," and "Circe" among many others.  More broadly, Milton is deliberately trying to write an epic poem, a form usually reserved for classical subjects, about an event that is central to Christian theology--the Fall of Man, and he is especially overt about this project in Book Nine.
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