I wouldn't wet the joint beforehand, it will stop the glue getting into the fibres properly.

Use about a foot of thread to get the glue right into the crack, put a generous bead of adhesive on the crack & use it like a cheese wire & saw it back & forth down the crack a few times then clamp it, I would use a decent sized G clamp with a couple of pieces of ply for padding. It would also be wise to fit the action before you clamp it to make sure everything remains in alignment, especially the fore end screws, I've found Airsporter stocks to be prone to some twisting if you give it the chance.
Wipe surplus adhesive off with a damp cloth then stick the clamped gun in the airing cupboard overnight.

I wouldn't use anything but waterproof PVA in a tight crack like that, I've not had much luck with superglues & other alternatives. Titebond is very good but I don't find it as strong as PVA - but if you ever have to do any joinery with wonky pieces of wood it's a Godsend.