you can call Chicago transit authority at 312 836 7000 and ask about bus schedules from union to midway the orange train line stops running at 11:30PM
I was looking for this myself. I found two buses, but no trains: Bus direct to Midway: Bus to Chicago's Union Station, and then take the Orange El train to Midway (longer): See the related links.
Midway Airport
Chicago Midway: IATA: MDW ICAO: KMDW
Midway International Airport is located in Chicago, Illinois. It is the second largest airport in the entire state of Illinois, after Chicago O'Hare International Airport, and is the largest airport in metropolitan Chicago.
The closest commercial airport to Chicago Union Station is Midway Airport.
Midway Airport is in Chicago; it's about 11 miles from the heart of downtown.
Chicago Midway International Airport.
CTA Orange Line
The driving time from Chicago Midway International Airport to Chicago O'Hare International Airport is about 41 minutes.
Chicago's O'Hare International Airport.
Chicago Midway International Airport current IATA code "MDW" .
"Also known as the Chicago Midway International airport, it is located approx. 15-20 minutes from down town Chicago. It is the second largest passenger airport in the Chicago area after O'Hare."