I think something along these lines would be suitable.
http://www.airgunsbbguns.com/Crosman_35 ... o3576w.htm
Things I would look for:
Pistol rather than rifle form factor, for ease of use close up.
CO2 powered, for more consistent results, and the ability to get a second or third shot off immediately.
.177 calibre, being the smallest available. Smaller calibres have better penetrative power, so I think this would be best.
I've owned two of these air pistols in the past. They had a slightly different appearance, being older models, but I can say it's a sturdy little gun. I'm not an expert on dispatching livestock though, so if anyone else has better ideas, please add them.