Not shoot with an arrow that's for sure.
LOL!! "Madam would prefer a 12 ga., perhaps?" LOL
IF the cat had been a problem, and IF there was no other viable alternative, I can understand the use of a crossbow or bow and arrow. Disharging a firearm might be illegal where this guy lives, or maybe he wouldn't want to atttract too much attention. But it is cruelty to let it suffer.
If on the other hand, this kid was simply up to mischief, particularly since the cat is someone's pet, he should be severely fined.
I vote for the alternative of using a sledgehammer or a nice rusty leghold trap.