"Do I measure the post near the bottom where it isn't 'tapered' in for the string?"
-yes
"Now what's an as-cheap method to measure the size of the posts?"
-measure the existing holes on the tuner bushings that you have now. try fractional drill bits since you are looking to purchase bushings for 1/4" dia. tuners.
you really should buy some calipers. they will open up new vistas of dimensional exploration for you for years to come
