There is a shuttle bus that will take you through the tunnel, but it is not a door-to-door service. Taxis go back and forth all the time. You could get one right at the hotel. When returning after the game, it would be best to have a pre-arranged ride. Taxis in Detroit are pretty much a call ahead service.

You definitely want to use the tunnel instead of the bridge. Much shorter route to Comerica Park, although downtown has a lot of construction right now, so it may look like you are going out of the way at times.

Crossing the border is usually not a problem, except if traffic is heavy it could take a while. Morning and afternoon rush hour are two of those times. US customs requests you have two pieces of photo ID to present them. I don't always get asked for two, but always have them just in case.