Socrates didn't say that, or rather we have no idea whether he really did. Plato writes that, in the Apology, and we have no idea whether he's actually reporting what Socrates said--the line is also more accurately translated "the unexamined life is unlivable," since adjectives like βιωτός indicate if something is possible, not if it's worth anything.

Orthography is a funny thing, not least because it's always changing and individual hands are...individual...Plato wrote not long after a transitional period in the Athenian alphabet, which is the only form of the Greek alphabet I'll discuss--there were many slightly (and not so slightly) different alphabets in different places). The Greek alphabet of later antiquity, which is what we write Greek in, was derived from the Attic script following the Euclidian reform of the alphabet in 403, which officially adopted the East Ionic alphabet. The East Ionic alphabet introduced to Attic, among other things, Ω, and changed H from an aspirate to the longer form of E. The latter change resulted in the need to add a symbol for the aspirate, for which the left half of H was used--we find this in some inscriptions and it eventually became the "rough breathing" diacritical mark we're so used to now. By and large, though, the official script of Athens was by Plato's lifetime more or less the same as the one we use now, with a few exceptions--notably, lowercase letters were unknown, as we're accents and diacriticals. But Plato's manuscripts would have been written in cursive, not the official script of inscriptions. The development of Greek cursive is not something that I'm very familiar with, and from what I know the early stages of its development are sort of in clear (could be wrong on that). Certainly in the early and middle 4th Century, when Plato was writing, cursive and so-called "semi-cursive" are sort of indistinguishable and overlap with each other in form and usage quite a lot. I'm not entirely sure what you mean by "genuine" script. What exactly Plato's manuscripts would have been written in, and what early copies would have looked like, is anybody's guess--we can conjecture based on what we know about the official and personal scripts at the time but of course we have no "original" or even early manuscripts. If you want it in Greek you might as well get it done in the standard script that we use in modern editions