This reminds me the older question Did Wittgenstein anticipate Gödel? There is more to it in the case of Kant than there was in the case of Wittgenstein though, at least in spirit. One could say that Kant pioneered in epistemology the stratification into levels of discourse which Gödel later applied to formal semantics.
When the Gödel theorem first appeared many mistook it for a paradox, like Russell's, a contradiction within a system of mathematics. Many included Russell himself, Wittgenstein and Zermelo, at least according to the traditional view, see however What sources discuss Russell's response to Gödel's incompleteness theorems? for a different view. The issue was that the paradox only arises if one mixes the levels of language. Gödel sentence is unprovable in the object language, the proof that it is nonetheless true is done in the meta language, if one properly distinguishes between the two the paradox disappears, and we uncover an interesting property of the object language. Russell, Wittgenstein and Zermelo were presumably thinking universalistically, within an all encompassing logical system.
What does this have to do with Kant? Kant also has a two level distinction, not in the language but in ontology, appearances and things in themselves. Like Gödel's Kant's predecessors were in the habit of instinctively identifying the two, antinomic reasoning was a direct result of taking this identification to its logical conclusion. What prevents Gödel sentence from being a paradox is a subtle rephrasing of "I am false" into "I am unprovable [in a language]". Kant similarly resolves the antinomies by relating them to appearances, our 'language of mind'. What creates the Liar is the language trying to handle unrestricted truth within itself, what creates the antinomies is mind trying to handle unrestricted reality within itself. Both paradoxes result from disregarding self-limitations, and are resolved by explicitly re-instating them. As long as we do not regard the "world" as both an appearance and a thing-in-itself there is no antinomy of it having and not having a beginning in time, as long as we do not regard true and provable as a single item there is no Liar. Taking the analogy further one could say that Kant would have regarded Frege's universalist logicism program as a logical case of transcendental illusion, reasoning about appearances as if they were things-in-themselves.