How often does the Sun pass through the spiral arm it was born in?

It never does because these arms are not solid and coherent objects. Every few million years as the arm advances, it involves a different collection of clouds gas and stars in the disk of the Milky Way. There are about 4-6 arms that cross the solar circle, and since the sun takes about 240 million years to orbit the center, this means that every 50 million years or so, the Sun enters a new arm system.