You can, sort of. You can take the Amtrak to Union Station, walk east .3 miles to the Quincy/Wells CTA, take the orange line to Midway (a 30 minute ride, an extra two bucks). All being said, you might be better off taking the shuttle from Bone Student Center (drops you off at the gate, 24 bucks or so, less than 3 hours). Good luck.
CTA Orange Line
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.
Midway Airport is in Chicago; it's about 11 miles from the heart of downtown.
Chicago Midway International Airport.
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."
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Midway Airport is approximately 10 miles from downtown Chicago.