How will retribution be exacted between people in worldly matters, such as one person causing another to lose a job opportunity, which resulted in the latter experiencing extreme worry, sorrow, and overwhelming distress, leading to an incurable illness, while he prays to God to restore his rights, by analogy with the retribution of the two rams?

The hadith you mentioned has some reservations regarding its chain of narrators, but a sahih (authentic) hadith with a similar meaning has been reported: "Indeed, rights will be given to their rightful owners on the Day of Resurrection, even the hornless sheep will be avenged on the horned sheep." So, the oppressed will have retribution taken from their oppressor in the Afterlife if the wrong was not redressed in this world or forgiven, whether the wrong pertains to money, honor, or anything else. Therefore, whoever wrongfully deprives a deserving person of a job opportunity has wronged them, and the owner of the lost right will have their right taken for them on the Day of Resurrection. This wrongdoing will be taken from the oppressor's good deeds. If they have no good deeds, then some of the oppressed person's bad deeds will be taken and cast upon the oppressor. The extent of all this is within Allah Almighty's knowledge. As for the distress and grief that this person experienced, it is recommended to turn away from worry and sadness, to be optimistic and hopeful, and to increase supplications to Allah Almighty, for perhaps the good was in the loss of that opportunity.