The Webley tri-pump I have sucks air in through one side of the tee-bar handle via a sintered metal filter and a pack of silica gel. Standing the pump in a bag would not be detrimental to the pump or pistol, but I can't see it making any difference...
As for the pump, I bought it for £50 from a mate as it had stopped working, I fixed it by replacing the tiny, tiny 'O' ring at the very centre of the pump. That has lasted two years, so I have stripped it down again, new seal with some cleaning and lubing, and hope it will be good for another good stretch.

Tanks are scary...