How does altitude affect drag?
Approximately, yes.
Both lift and drag depend on the density of air, so in order to generate enough lift at the same angle of attack in thinner air, an aircraft has to fly faster. Thus the same number of air molecules hit the plane every second and the drag is approximately the same.
This is a bit of a simplification (especially the bit about drag being caused by air molecules hitting the plane) as the change in temperature causes a slight change in viscosity, which affects one type of drag (detailed Peter's linked question). However it's a good enough approximation for most pilots, who use 'Indicated Air Speed' as a air-density-independent guide to aerodynamic speed.