Thanks for the tip on the Groupon, I got some passes for my trip too. The apartment we're staying in has parking included and I'm just as happy parking the car and leaving it for the 4 days we're there. Especially after driving from Austin!
After hours of Internet searching different properties, i found a 3 BR place on vrbo.com.
It is in an old Swedish neighborhood, Andersonville. 4 nights, with all charges, comes to $781. This includes one garaged parking space.
37 reviews are all positive, including many comments on how easy it is to take bus downtown and to Wrigley Field and how the Andersonville community is fun after a day in the city.
I feel really good about it.