Is it ok to shoot an air gun (also known as a "BB gun") to knock over cups on Shabbos?
I can't think of any reason to prohibit it.
while an air rifle may not have the qualities that a real firearm possess in function it can still be used for the same purpose. What I mean is an air rifle can and many are used for hunting small game such as birds, squirrels, and rabbits. granted someone who is keeping shabbos is probably not going to be hunting the fact that it can be used for things not permitted on shabbos would make it muksa. To make a comparison even if one doesn't normally watch tv during the week the tv remote at someone elses house would still be muksa even for you. In addition o the issue of muksa I don't see using an air rifle as an activity fitting for shabbos
Isn't there an issue with "Activities that are not appropriate for Shabbat, even though they are technically permissible according to the Torah (uvda d'chol)."?
It can be a problem of Zorah (winnowing) because of the thrust of air.
Might be muktzah depending on the model. There are airsoft guns made to resemble specific real weapons as closely as possible for training. One of those might be assur to use for reasons of uvdin d'chol but I can't think of a reason to forbid a toy BB gun.
It's fine because no melocho is involved and it's not winnowing because the bb and the gun were never one entity to begin with.