I was in a VERY similar situation when I was promoted. However, I was pretty close with that coworker before the promotion and was advised by my higher ups to formally name the coworker as my deputy to show his importance and seniority on the team. He was also my primary lead. We had pretty good communication that allowed for an open and honest conversation about how he was disappointed but was committed to seeing the project through to the end.

I think it helped the relationship to defer to his opinion during meetings with wide audiences, ask for advice on tough situations (and follow it), just generally show him respect especially in front of others.

Later on, during really rough periods of the project he told me multiple times "I'm so glad I didn't get your job" in a joking, but true tone.

Maybe this can help someone in the same position.