Thank you for sharing honestly about what you are facing at work. It makes sense that you would feel dread about spending a whole shift one on one with someone who has been diminishing your work and trying to correct you in front of management. Wanting prayer for peace and for that competitive spirit to be calmed is a right and good request, and I am praying with you for that time together.
What you are describing has a lot to do with jealousy. In human terms jealousy usually comes when we feel our territory is threatened. It is often rooted in selfishness, and Scripture lists it among the works of the flesh in Galatians 5. That is very different from the way the Bible speaks of God being jealous. His jealousy is holy. It comes from love. He is jealous for you because He loves you and wants what is best for you, knowing that anything that pulls your heart away from Him will ultimately leave you with less than His best. Your coworker's response to your promotion sounds like that human kind of jealousy, the competitive, territorial kind that wants to prove itself by making someone else look smaller.
It helps to get a longer view when we are in the middle of that kind of tension. One psalmist admitted he was envious when he saw the ungodly prospering, until he came into the presence of God and his vision was corrected. He moved from a close up, short term perspective to seeing things as God sees them. When you bring this situation before the Lord, He can give you that same correction of vision. You can remember that God sees your work, He knows your heart, and your value is not determined by whether a coworker acknowledges it.
Scripture also reminds us that vengeance belongs to the Lord. He says, vengeance is mine, I will repay. There will be a day when all things are made right, and you do not have to sin by trying to settle the score yourself or live with constant defensiveness. You can entrust both yourself and this coworker to God, who judges justly. That does not mean you accept disrespect without wisdom, but it does mean you can release the need to prove yourself or to put him in his place.
We live in a world full of conflict, between groups, in workplaces, and often within our own hearts as the flesh struggles against the Spirit. Walking with Jesus will sometimes put us out of step with the way the world handles conflict. The good news is that the gospel brings peace, first peace with God through Jesus, and then peace in our own hearts that can steady us even when others are stirring up strife.
The apostle Paul faced something similar. While he was in chains, others who were jealous of what God was doing through him tried to use his situation to advance themselves out of a spirit of contention. Paul did not respond with jealousy or competition in return. Instead he rejoiced that Christ was being proclaimed and refused to be drawn into that rivalry. That is a helpful model for you. Rather than feeling you must compete back or constantly defend yourself when he corrects you, you can choose to rejoice that God is at work in your life, do your work faithfully and quietly, and leave the outcome in God's hands.
For your shift together, pray before you go in and ask God to fill you with His peace and self control. Determine ahead of time to stay calm, speak clearly and respectfully, and not be pulled into an argument. If he corrects you unnecessarily, you can respond simply and factually without attacking back. If he tries to diminish your work, continue to do what is right and let your consistent faithfulness speak. Ask God to guard your heart from resentment and from a competitive spirit in return.
I am praying for you, that God will calm anxious thoughts as you go into this time, that He will quiet the competitive and bossy impulses in your coworker, and that He will give you wisdom, grace, and peace that others can see. May He take away that spirit of jealousy and replace it with a more humble and respectful attitude, and may He help you to rest in the knowledge that He sees you, He is for you, and He will care for you through this.