This link should work better: Union Station Multiplex to Chicago Midway International Airport - Google Maps
I think you meant Union Station to Midway Airport. Since it'll take 45 mins by train you may want to consider a car rental place closer to Union Station.
That is in the business district and I think all the national car rental places are near the hotels East northeast of there. And within 15-20 min walk from Union Station.
If you rent a car from an airport you usually have to pay airport facility fee and other taxes that you aren't charged when the car rental place is off the airport property.
Check your available discounts or try a travel site like Travelocity.com