Does storing a hygroscopic reagent in a bottle makes it prone to absorb humidity from the air trapped inside?
Yes, there is that tendency. I'll make a couple of points which I'll number for possible further discussion.
(1) 99.5% acetone is probably 0.5% water. So the "extra" water from the air above the acetone in the bottle (from one opening and closing of the cap) would be negligible compared to the amount of water already in the acetone. Obviously when using the acetone, uncap it and recap it ASAP to keep it as dry as possible. The "problem" is opening and closing the bottle many times.
(2) I looked quickly at Wikipedia, and do a quick a google search and didn't find any water-acetone azeotrope. So the distillate would be "pure".
(3) I'm not sure how pure a simple distillation in air of the acetone would be in removing water. I'd guess that the resulting purity would be like your 99.5% starting purity.
(4) Vapor Pressure would play a role in that the acetone would have a partial pressure in the bottle. So not all of the void in the bottle would be air. Acetone doesn't "boil" at room temperature though so the partial pressure won't be enough to keep some air from seeping in and out of the bottle. In other words the void in the bottle is mostly air with just some acetone.
(5) The !@#$%^&*( bottles "breath" as the air pressure changes if they are not absolutely sealed.