The correct answer is: A. have become
The sentence is in the future perfect tense, which is used to talk about something that will have happened by a specific time in the future. In this case, the speaker is saying that they expect the listener to have become bored with their job at the Post Office by this time next year.
The other options are incorrect because they do not express the future perfect tense. B. will have looked is in the future perfect progressive tense, which is used to talk about something that will have been happening for a period of time by a specific time in the future. C. were becoming is in the past progressive tense, which is used to talk about something that was happening at a specific time in the past. D. are going to look is in the simple future tense, which is used to talk about something that will happen in the future. E. will have become is also in the future perfect tense, but it is in the passive voice. The passive voice is used when the focus of the sentence is on the action, not on the person or thing doing the action. In this case, the speaker is not saying that the listener will have become bored with their job, but that they will have been bored with their job.
Here is a diagram that shows the difference between the future perfect tense and the future perfect progressive tense:
Future Perfect Tense | Future Perfect Progressive Tense
———-|———
Action is completed by a specific time in the future | Action is happening for a period of time by a specific time in the future
Focus is on the action | Focus is on the action and the time it takes to happen
I hope this explanation is helpful!