I asked a knowledgeable local beer list about this and I'll put responses here as they come in:

1. "There's a Reilley's Daughter inside the terminal "food court" that at least has
Guinness, Smithwick's, and maybe Magner's/Bulmer's cider. For the moment there's a lot of Stella on tap at other stop-ins by the departure gates. The Harry Caray's outpost there MIGHT have the occasional Goose Island [a good local brewpub's brand]. But outside of the airport, the closest I can think of would have to be the Maple Tree Inn in Blue Island or BJ McMahon's on 95th in Evergreen Park."

toomanybook's comment: Those are pretty far away, unusable unless you have a 5 hour layover or something. Midway is at about 57th, so 95th is about 5 miles away minimum. There would be some local places down there, though, including some good pizzerias. Some of the bars in the MDW area look to me from the outside as though non-regulars might not be too welcome. I will be driving that way early next week, and I'll try to keep my eye out for possibilities.

2. There's a Vaughn's Pub on 65th and Central. They have probably the best
selection in that area. If you want some southside ambience and decent beer, Mr.
C's on 63rd and Kilpatrick is a pretty friendly place.

BJ McMahon's is on 95th and 54th Ave in Oak Lawn, and has the best selection
in that area.