When Farrington is publicly reprimanded, it is at the end of the day. Everyone is a bit frayed and there is an air of tension present. Farrington's mid day drink as well as his desire to continue drinking don't help the situation. Additionally, everyone in the office, including Miss Delacour, are now watching the power play between both manager and worker. Farrington's first reaction is to deny that he knew nothing of the two missing letters. He lies, hoping that Alleyne would not notice their absence. Indeed, this calculation backfires and his boss does notice the missing items. When Alleyne asks Farrington if "he takes him for a fool," Farrington responds with his fateful wise crack": "I don't think, sir," he said, "that that's a fair question to put to me." Even Miss Delacour grins because, in the context, it was a fairly challenging comment to make. This only fuels Alleyne's anger, which reaches epic proportions in what he will do to Farrington.
In terms of assessing whether his reaction is justified, opinions will vary. Certainly, those who criticize Farrington's actions will suggest that he knew he was in the wrong with not completing his job of copying the two additional letters and wanted to get away with it. They will also point out that the price of seeking to evade work is getting publicly reprimanded. Additionally, given Farrington's domestic responsibilities with wife and children, he needs to keep the job to support them. In this light, Farrington was not justified in "mouthing off" to his superiors.
The case for him being justified would rest with mutual respect. Mr. Alleyne had not been professional with Farrington throughout the narrative. The public reprimand was the final straw. Essentially, Farrington was put in a power situation and flipped it, if only for a moment. If he is seen as justified, it would rest with the idea that Farrington had been pushed to a brink, to a particular point, and he responded with what he could. Certainly, his flippant response is much better than the violent rage that is present within his psyche in the narrative. The more appropriate question regarding Farrington's appropriateness would be more applicable with how he beats his son at the end of the narrative. I think that the question of justification lies here and not with his boss.