While not a big deal, in English we don't use the noun "cup" with alcohol. Asking for a cup of beer or a cup of wine sounds a little bit strange.
Instead of using "cup", ask for a glass, pint, or bottle of beer, depending on what your preference is.
"I want a cup of beer please."
"I will have a cup of wine please."
"I'll have a cup of draft beer please."
"I'll have a glass of beer please."
I'll have a glass of wine please."
I'll have a pint of draft beer please."
1.) Use 'cup' for tea or coffee. Use glass for everything else (juice, milk, water).
2.) A pint is 16 ounces of beer. We usually order a "pint of draft" from the bar. "Draft beer" is not from a bottle but from a keg and is poured directly into a pint glass from the tap.
1.) "Let's go out for a drink": Is a common phrase to invite someone to have a drink with you. "Let's go out for a beer" is much more natural than saying: "Let's drink a beer in a bar tonight."
"Hey, if you're not busy tonight, how about we go out for a drink tonight?"
2.) "This place is dead.": Is an expression used to describe a bar, restaurant, or club that is empty and has no people.
"Let's go to the bar after midnight. It's usually dead until around midnight anyways."