The Arch is in the middle of a nice park, so if it's decent out, that's a great place to go during lunch or just to enjoy the sun for a bit. Same goes with CityGarden, which is a few blocks away (across the street from where I worked when I still had to commute into the office--look for the 30-some story smoked-glass building); it just opened last spring-ish, if I'm remembering correctly. Kiener Plaza is also a nice spot to sit and eat lunch if the weather's good (and it's impossible and completely futile to predict if it will be. It could be 80 degrees or it could snow. It snowed a bit for the Cards' home opener last year.)
(Kiener Plaza)
The problem is the lack of car. While they are "revitalizing" the downtown area, there's so much more to do that's within about 15 minutes (by car) of downtown (Art Museum, Science Center, Zoo, etc.)
The Cards will still be in FL for Spring Training, so that's out. But the stadium is a nice walk away from the convention center. They play classic game audio over some speakers during the day (I assume they still do...it's been a while since I was there) that's audible outside, and the team stores and stuff are open. Oh, and they've got marked on the sidewalks outside where the outfield fence and one of the foul lines were in Old Busch, which is cool if you're a baseball dork like me.
The tacky bomb-crater-lookin' thing and construction equipment are no longer there. Blessedly, they got all of that cleaned up before the All-Star Game last summer.
Please excuse the awful cell phone pics.