There is no public transportation system in Morehead City; you'll need a car to get much of anywhere, although there are a number of reliable taxi companies in the area. The main downtown thoroughfare is Arendell Street pronounced "uh-REN-dull"; IPA [(IPA: É.'rIPA: ʌn.dIPA: Él]).
The bus station is located at 105 N 13th Street +1-252-726-3029 if you want to catch a ride to other towns in the area; you'll need to call the Carteret County Area Transportation System (http://www.co.carteret.nc...) at 252-240-1043 for a reservation ahead of time. Tickets start at $2.50.