At night, he goes out to grab a drink.

Literal

Night, he [topic-は] outside [to-に] go-out-[and-て] one-cup do.

一杯やる ('have a drink,' lit. 'do a cup') is a casual expression for drinking alcohol socially. 一杯 ('one cup/glass') takes on the idiomatic meaning of 'a drink' in this context. ~て chains two clauses sequentially.