If you do not have or hire a car you will be reliant on taxi or airport shuttle. I don't hire cars after a long day's travelling across the Atlantic (not safe imo) so I chose to use the Airport Shuttle Service advertised on the airport's own website.
It is not possible to book using the online website form if you live outside the US (unless you happen to have a US phone number) . I booked by telephone, friends booked by email. Both worked perfectly.
Airport staff were very courteous and pleasant, as were the drivers. A variety of vehicles are used, depending on numbers (it's a shared shuttle service): I unfortunately arrived during 'moving-in week' for Iowa University, so had to share my ride with several students and their very large bags. My friends were much luckier, having the ride to themselves.
I'd highly recommend this company if you need onward transport from the airport.
Cedar Rapids does have its own airport with regular flights by major airlines, but the fares are high. I've found that the fares are significantly lower to Moline IL which is about 1:15min drive. I'm a cheapskate and find an hour drive is no big deal, so I usually fly into Moline.
You must have your own car to get around in Cedar Rapids. Unless you're staying downtown, there are a few things you can do within walking distance, but your opportunities are limited.You will not be able to just hail a cab or find them parked and waiting anywhere, you have to call for one and be prepared to wait 20-30min.