The are 4 different ways across 3 unique modes of transportation to help you travel from Iowa City to Minneapolis. The fastest way out of Iowa City towards Saint Paul is by bus, as it is the quickest to the Minneapolis area. This doesn't necessarily take into consideration traffic or weather - so these may impact some of your decision making. It's common to find tickets for as low as $44. United operates from Iowa City to Minneapolis, so you book your trip with them! Book as soon as you can, as there may be limited availability.
On your trip from Iowa City to Minneapolis, it is best to book as soon as possible. It won't necessarily help reduce your price by a large amount, but will ensure that you have a spot on the bus or train.