No, you can't. 'Compensate' is a common word in this context and it refers to something practical: it just means that they paid you for the time you worked or gave you extra time off.
'Atone' is not a normal everyday word. It has moral and religious overtones and we would never use it in this context. Atonement relates to the moral concept of wrongdoing, and is something personal to you, or between you and God.