Unless you've brought a bicycle and happen to be in good shape, you aren't getting anywhere in Minot without a car. Some public transit is available at certain times, but for visitors it's more trouble than it's worth.
Car Rental
Avis 838-7665, Hertz 852-0104, and National 852-5115 located at the airport. There is a Rent-a-Wreck 838-0098 at a Sinclair gas station three hilly blocks south of the Amtrak station. There is an Enterprise location 838-3800 on the south end of town. Enterprise and Rent-a-Wreck will pick you up; Avis, Hertz, and National are oriented primarily towards airline travelers.