Because the Omer count runs forty-nine days, the festival falls on the fiftieth. Shavuot in Judaism marks the wheat harvest and, in rabbinic tradition, the giving of the Torah at Sinai. The Greek-speaking Jewish world called it pentekoste, simply the fiftieth, and that name carried into Christianity as Pentecost, where Acts 2 places the descent of the Spirit on the apostles on the same day. The Christian feast therefore falls where it does because of an agricultural counting rule in Leviticus, which is a clearer line of descent than most liturgical dates enjoy.
Fifty appears elsewhere in scripture as a round administrative number, captains of fifty, fifty-cubit measurements in the tabernacle and temple accounts, and as the number of days the flood waters prevail in one reckoning.
The fifty gates of understanding are more interesting and are frequently mis-sourced. The saying appears in the Babylonian Talmud at Rosh Hashanah 21b: fifty gates of understanding were created in the world, and all were given to Moses except one. The proof text is Psalm 8:6, that he was made a little less than God. This is Talmudic, redacted around the fifth or sixth century, and not, as often claimed, a Kabbalistic invention.
What is later is the elaboration. The phrase Nun Sha'arei Binah and the reading of the fifty gates as levels within the sefirah of Binah belong to medieval Kabbalah and to the Lurianic system of the sixteenth century, where they are connected to the forty-nine days of the Omer as an ascent, one gate per day, with the fiftieth reached only at Sinai. The Talmudic line is old; the architecture built on it is not.