Geodesics on the sphere are great circles, and a great circle is the intersection of a plane which passes through the centre of the sphere with the surface of the sphere.
In the general case (i.e. the random angle isn't a multiple of $90^\circ$), the man will cross the equator on the antipode of his starting point, because the intersection of the plane in which the equator lies with the plane of the circle he follows is a line passing through the centre of the sphere.
Suppose that his angle is between $0$ and $180^\circ$. Then he starts by going south of the equator, reaches a furthest point south after a quarter of his journey, carries on moving back to the equator and reaching it at the halfway point, etc. So for any latitude between the equator and the southmost point, he passes through two points at that latitude. This is why we need to know the starting point: without that, we don't know whether the current point is the first or the second point at that latitude which he reaches.
Given the start point and the angle, you can parameterise the great circle, and then given the current point you can calculate the current value of that parameter with arc-tan and quadrant considerations. You can also calculate the great circle distance between the start point and the current point by taking the arc-cos of their dot product, and you can extrapolate using SLERP.