According to the Talmud, Pharaoh's dreams and the release of Joseph from prison (see Genesis 41) occurred on Rosh Hashanah, the first day of the year. On that day, G‑d determines the fate of the world for the coming year and, among other determinations, decrees whether it will be a time of famine or plenty. It is also the day when G‑d decides the fate of every individual for the upcoming year.

It was therefore on this day that Pharaoh dreamed about the coming years of plenty which would be followed by the years of hunger. And it was on this date that G‑d decided that the time had arrived for Joseph to become a free man and be elevated to royalty.

Another notable event that occurred on Rosh Hashanah – for the same reasons mentioned above – was the cessation of slavery for the Israelites in Egypt. Though the Exodus occurred in the spring, in the month of Nissan, active servitude on the part of the Israelites came to a halt more than a half a year beforehand—on Rosh Hashanah.

Let me know if this is helpful, and if there is anything else that I can assist with.

All the best,

Rochel Chein for Chabad.org

Source:
Talmud, Tractate Rosh Hashanah 10b.