The Greek reads καὶ εἶπεν αὐτοῖς ὁ Ἰησοῦς· τὴν ἀρχὴν ὅ τι καὶ λαλῶ ὑμῖν, which has ἀρχή ("beginning") in the accusative case. The Latin principium is a 2nd declension neuter noun that could be either accusative or nominative.
In fact, neither conveys what appears in most English translations (... from the beginning). The phrase "from the beginning" would have been indicated by ἀπʼ ἀρχῆς (e.g. Matt 24:21) or ἐξ ἀρχῆς (e.g. John 6:64), which take ἀρχή in the genitive and not the accusative case and require a preposition.
So in answer to your question, I would say that yes, the Vulgate is accurately translating the Greek phrase (Latin accusative, it seems, for Greek accusative). But the Greek itself seems somewhat ambiguous.
Metzger's Textual Commentary on the New Testament offers the following additional insights:
Since the older Greek manuscripts lack punctuation and are written without division between words, it is possible to interpret Τὴν ἀρχὴν … ὑμῖν in several ways:
(1) As a question, with ὅτι = why? (“Why do I speak to you at all?”).
(2) As an exclamation, with ὅ τι in the sense of the Hebrew מָה (“That I speak to you at all!”).
(3) As an affirmation, with ὅ τι and supplying ἐγώ εἰμι (“[I am] from the beginning what I am telling you” or “Primarily [I am] what I am telling you” or “[I am] what I have told you from the beginning”).
Several Latin witnesses (and the Gothic), misunderstanding the Greek, translate Principium, qui et loquor vobis (“[I am] the Beginning, even I who speak to you”). The Ethiopic omits ὅτι (“[I am] the Beginning, and I told you so”). The Bodmer Papyrus II (𝔓66) reads, according to a marginal correction that may be by the original scribe, Εἶπεν αὐτοῖς ὁ Ἰησοῦς, Εἶπον ὑμῖν τὴν ἀρχὴν ὅ τι καὶ λαλῶ ὑμῖν (“Jesus said to them, I told you at the beginning what I am also telling you [now]”).
The Metzger commentary further notes:
For full discussions of the difficulties of the passage, see R. W. Funk, Harvard Theological Review, LI (1958), pp. 95–100, and E. R. Smothers, S.J., ibid., pp. 111–122, who independently prefer the reading of 𝔓66c.