What is the ruling regarding a mother's anger towards her son if he wishes to perform a necessary act that she disapproves of, and must the son abandon this act?

If your mother has an interest in you leaving your job, and you would not be harmed by leaving it, then you must leave it in obedience to her. However, if it is not obligatory, then you should appease your mother by explaining your need for the job and that she has no interest in you leaving it, and that obedience in such a matter is not obligatory. If you can leave it to gladden her heart, then that is an act of righteousness.