As far as the out-state citizens are concerned, I don't think they have anything to worry about. They too will see the benefits. They cannot argue that they don't help out the Minneapolis economy looking at all the trips to the cities people around the state make per year. Many people around the state make trips down to the cities just to watch the Vikings, Twins, or Timberwolves play. The state is the one that is recognized when the team is in the World Series or the Super Bowl, not just the city.
Professional sports does just that.
Therefore, it shouldn't matter to the out-state residents if the money is going to better their local community or the state community. People from all over the state are going to want to see the new stadium. Even people from other states will come to games just for the sake of checking out the new facility. It gives them more of an incentive to attend the game, even if the ticket prices are higher. You can't tell me that if we were to build a new stadium for the Twins or the Vikings or both that you would not want to go to a game just to see what it is like.