I have more years of work experience than my boss has been alive. But he is one of the best bosses I've ever worked for. First, he has made it clear he values my experience and he asks for my input into a variety of technical issues. But it is also clear that while he listens to what I have to say, he is the final arbiter of what we will do. Second, he rewards good work and lets people know when they need to improve - his subordinates are never kept guessing as to how he feels about their work. He is fair. Nobody is perfect and he will need to discuss issues with any one of us. He discusses issues in private and gives out praise in public. Issues when they are brought up are never personal attacks or blame sessions. The discussions are more along the lines of this is what went wrong and what can we do to prevent it from happening again. He seeks input and more importantly, he makes sure to take that input to higher levels and makes sure we get the credit for our suggestions if higher management buys off on them. He pushes for promotions and pay raises for his people. We have an awards program and he takes the time to make sure at least one of his employess is nominated each time the awards cycle comes around. He makes it easy to tell him you need emergency leave and to make sure your work is covered. It is easy to accept someone with less experience as your boss when he treats you and all his other subordinates well.
Now it's trickier if the person wanted that promotion himself. It's best to sit down and talk about the situation openly. Tell him how much you value his experience; tell him that you want to make sure the next promotion has his name on it. If you know why he was passed over (sometimes it's obvious to everyone except the person passed over), then discuss that too. Let him know what he needs to do to improve his chances of being selected. Likely he is not as politically savvy as he should be, help him to see what actions he needs to take to get credit for his work and if you promise to help promote his cause to management, make sure he hears you giving him credit for his good work. One compliment to a higher boss that he hears will do a lot ot relieve any anxiousness about you.
That said, you are the boss now and he has to respect that too whether he is happy about your promotion or not. So don't let him get away with behaving badly. The sooner you confront behavior like trying to override what you are saying in public, the sooner it will get fixed. If you don't confront early, it may very well escalate until you have an actual performance issue to deal with officially. If he thinks he call bully you, he will. Tell him privately that you don't want him to interrupt you when you are talking to others. Tell him you will ask for his input when training others or presenting things to management and make sure he gets the credit.