Acetone absorbing water from the air
For Acetone or IPA we are more than happy to keep it in normal squeeze bottles which are not air tight so the bottle you've got should be fine. I believe you're more likely to see it evaporate first than to see it get 'noticeably weaker' (I assume you're using technical grade acetone since it's just for cleaning). There should be no need to dump the acetone/IPA in the bottles on a weekly basis - although acetone is cheap its unnecessary to waste money like that!

Along the lines of trying to figure out what could be the cause of the experience of your staff, I doubt it has anything to do with the acetone itself because you say that they can use the same chemical on two different days and have two different experiences. So it's either they're imagining it (let's assume they are not), or it is the dirt that has to be cleaned that is the issue.

I've experienced it in the lab before that sometimes it does seem more difficult to wash out and clean glassware despite using the same chemicals - and this usually has something to do with how long the 'dirt' has been sitting in my flask or how it got there in the first place (slow precipitation/crystallization, crashed out, solvent dried out and left overnight, etc.). Without knowing what your product is and what it is you're trying to clean out, I can't offer much else other than checking out the consistency of the incoming product that needs to be cleaned. It could be a sign that the manufacturer's process is contaminated on some days.