You can easily walk to Busch Stadium if you want, and, in fact, that might be easier than trying to board the train that close to the ballpark; it is usually full headed to the games, and VERY full leaving the games.
As mentioned, Union Station is a spot to kill a little time. Also, if you're a gambler, take Metrolink to the Laclede's Landing stop (another place to visit with tons of restaurants in the area), or to the East Riverfront stop. There are casinos within walking distance from both of those stops.
The Redbird Express is indeed a special shuttle that runs to/from Cardinals games; you wouldn't be taking that. As previously mentioned, depending on how many ons and offs you're going to do, the one day pass might be worth it. A single ticket is good for two hours in ONE direction of travel, so, you're not exactly allowed to use it for a round trip. That being said, I don't know how closely the tickets are checked, regarding the validation times on them.
Metrolink does indeed go to the airport, as also previously mentioned. Here is a link to the Metrolink site, so you can get a little more comfortable with it before you get to town:
Metrolink.
Enjoy St Louis!!