What is the idiom that best describes someone doing his job carelessly as if he looks like fed up with his job or needs to get fired?
"To slack off", meaning to work less hard than is usual or necessary, is another possibility. Although Cambridge lists it as American English, I have little doubt it is well understood by global English speakers. "Slack" can also be used by itself, as for example in "he's so slack at work it's a wonder he hasn't been fired yet".
Other useful words, probably more standard English than idiomatic, include