The traffic to and from Iowa City is always minimal so even the distant airports are reasonable travel times.
The closest and best option is CID: The Eastern Iowa Airport (Cedar Rapids-Iowa City): Distance 22 miles (25 minutes)
United, Delta, American, Allegiant , Frontier
Additional options are listed below
MLI: Quad Cities Airport (Moline, IL): 64 miles (1 hour 5 minutes)
Airlines: United, Delta, American
DSM: Des Moines, Iowa: 126 miles (1 hour 45 minutes)
United, Delta, American, US Airways, Frontier, Southwest, Allegiant
PIA: Peoria, Illinois: 150 miles (2 hours 20 minutes)
Allegiant, United, Delta, American
ALO: Waterloo, Iowa: 88 miles (1 hour 20 minutes)
DBQ: Dubuque, Iowa: 97 miles (1 hour 35 minutes)
Chicago is a major hub and it may be a reasonable option if you want to rent a car. The drive is very easy from both Midway and O’Hare.
ORD: O’Hare: 230 miles (3 hours 15 minutes)
MDW: Midway: 236 miles (3 hours 30 minutes)