how many angles do you need to measure to map out the position of a sky object at any particular time?

Answer

Two angles - altitude and azimuth

Q: What are the angles used to indicate the position of celestial objects? A: The two angles used to describe the position of a celestial object are altitude and azimuth.
Q: What is altitude and azimuth? A: Altitude is the angle between an object in the sky and the horizon. Azimuth is the angle between the object and due north, measured along the horizon.