For our purposes, the important point is that the archetypes are essentially the Platonic Ideas.  This, in itself, is not a new notion, nor is it surprising.  Indeed, Jung (CW 9, pt. 1, ¶5) says, “‘Archetype’ is an explanatory paraphrase of the Platonic εἶδος,” and he cites its use by Philo Judaeus (De opf. mundi, I.69), Irenaeus (Adv. haer., II.7.5), the Corpus Hermeticum (I.8, II.12), and pseudo-Dionysius (De cael. hier., II.4; De div. nom., I.6); the term is also used by Plotinus (e.g., 5.1.4).  Indeed Jung (CW 8, ¶154) defines archetypes as

active living dispositions, ideas in the Platonic sense, that preform and continually influence our thoughts and feelings and actions.

Certainly Jung seems to have been influenced more directly by Gnosticism than by Neoplatonism, for the Gnostics, as Jung (1965, 200) says,

had been confronted with the primal world of the unconscious and had dealt with its contents, with images that were obviously contaminated with the world of instinct.

But Gnosticism itself has many connections with Middle Platonism and Neoplatonism.  Eventually Jung became interested in alchemy because he saw it as a “bridge that led from Gnosticism — or neo-Platonism — to the contemporary world” (op. cit., 201).

As psychologist James Hillman (1975a, 198) remarks, “There are striking likenesses between the main themes of Neoplatonism and archetypal psychology.”  He notes (Hillman 1975b) that although Jung cites Neoplatonists infrequently, he was inspired at an early stage of his career by the Neoplatonist scholar Friedrich Creuzer, who later edited the works of Plotinus, Proclus, and Olympiodorus.  Jung (1965, 162) says that he “read like mad” Creuzer’s Symbolik und Mythologie der alten Völker, and “worked with feverish interest” through this Neoplatonic analysis of mythology.  Hillman refrains from claiming a direct dependence of Jung on Plotinus via Creuzer, but he does

want to suggest, and strongly, that the reason Jung was so fired by Creuzer was because he and Creuzer shared the same spirit, a profoundly similar psychological attitude, an archetypal attitude, which tradition calls Neoplatonist (Hillman 1975b, 149). 

He notes further affinities, calling the Florentine Neoplatonist and theurgist Marsilio Ficino the Renaissance patron of archetypal psychology (1975a, 200), and claiming, “Ficino was writing, not philosophy as has always been supposed, but an archetypal psychology” (1975a, 202).  (See also Hillman 1975b on Ficino.)

In summary, there is reason to conclude that Jung was influenced by Neoplatonism both directly and indirectly (via Gnosticism and alchemy), but even were he not, we can see the connections now and use each to illuminate the other.