Are υπάρχει and είναι used the same way in Modern Greek?

Not quite.

In the existential sense that you’re using, είναι ένας άνθρωπος means “it is a man”, and υπάρχει ένας άνθρωπος means “there exists a man”. The latter sounds as formal and logical in Greek as it does in English, though I think it is more widely used than English as an interrogative or negative. The former is pretty much an answer to a question, and presupposes that the context is known; for example, it would make sense as the answer to “who’s there?”

(Oh, and using the indefinite article is always a red alert for translationese: it really doesn’t get used that much in Greek.)

The idiomatic rendering of “there’s a man”, introducing their existence out of the blue, is neither: it’s έχει [έναν] άνθρωπο, “it has a man” (cf. French il y a, and Portuguese tem). So when a diver first saw the shipwreck with the Antikythera mechanism, he exclaimed: κάτω έχει ανθρώπους και ζώα σπαρμένους, “there are men and animals sown [= scattered] down there”: «Κάτω έχει ανθρώπους… σπαρμένους».

In formal Greek, you can use υπάρχω with a predicate, to indicate a person’s previous appointment: Ο κ. Πετσάλνικος που υπήρξε αρμόδιος υπουργός όταν ξεκίνησε η προδικασία της υπόθεσης “Mr Petsalnikos, who “existed” [served as] the responsible minister at the beginning of the litigation” (Ο Καλατράβα του Κορυδαλλού). But that’s journalese, not natural Greek: υπάρχω does not normally take predicates.