New United boss Louis van Gaal confirmed last week that Rooney will wear the armband this season after former captain Nemanja Vidic moved to Inter Milan.
In his Daily Telegraph column, Neville said: "There has been a sharp decline in leadership in football but Rooney has all the attributes.
"Rooney is one of a dying breed of footballers who is willing to call his team-mates out. He can take criticism but also hand it out.
"Van Gaal will have looked at Rooney and thought: this is a player who is not too proud to listen to criticism and not too shy to dish it out.
"That is captaincy material, as Van Gaal outlined in his press conference ahead of the Swansea match. The next stage for Rooney is to think like a captain. When I think back to his early days at United I remember his personality and cheekiness.
"Rooney was the only player who would dare pull up a seat at the coaches' table in the canteen and ask Sir Alex Ferguson: 'Am I playing tomorrow, boss?' The manager would cuff him about the head and say: 'That's nothing to do with you'."
Rooney told United Uncovered: "It's a massive role to take on - I'm aware of that, the responsibility of it - and I feel I'm ready at this stage in my career to do it.
"I spoke to the manager and he decided that I was right to take the role which I am delighted with. It's a great honour."
The 28-year-old tweeted: "Very proud to be named captain of @ManUtd. Huge honour and I can't wait to lead the team out on Saturday for our first game of the season."