"Can I get one cup?"
In reality, placing the counter first actually emphasizes the number of items. You could say both nihon no biru o nonda, and biru o nihon nonda, with the first one emphasizing the fact that you drank two bottles, whereas the second one just means you drank beer - two bottles of it.
カップ is cup, as in 'coffee-/teacup', and also for 'trophy' and 'bra size' kind of cups.
コップ is glass, though (like in English) it does not always have to be made of actual glass; ceramics and plastics may be referred to by this word as well.
Sidenote: グラス (from the Dutch "glas") is 'glass' - the material. So for wine/whisky glasses and such, you use that word.