a 12 foot pound (fpe) .22 cal air rifle will take down a rabbit at 50 yards with a headshot.
Just don't get too cheap thinking any old plinker will perform like a world-class air rifle.
There are videos of people on YT with $100 air rifles talking about how awesome they are, and that makes me cringe, because it's clear they have no experience with air rifles beyond their "wallmart special".
The cheapest i'd feel comfortable recommending would be a Walther Terrus ($200-250 on
pyramidair.com, depending on the stock option).
Of course, if you fancy spending more, you an get much better performance. We're talking around $1k for a top end springer and some good glass to go with it, and for a pcp, the price can easily exceed several thousand (i'm talking top end competition guns with all the bells and whistles).
Pic related is one of my springers (AA tx200hc .22 with a Hawke Sidewinder 10x42 1/2 mil-dot scope).