Cask ale is only carbonated to about 1-1.5 volumes, so even if it's flat, you wont notice a huge difference. While a cask does allow a little bit of oxygen, the natural co2 that the beer produces usually offsets this. Cask ale is usually put into the barrel/keg before it's fully fermented. A hand pump like the one you mention actually forces air into the beer, so it goes bad much faster. Still...I wouldn't personally drink a cask ale that's more than a week old.
Is this bar actually serving real cask ale, or just normal kegs hooked up to a beer engine?