Miyazaki Airport is located in the southern part of Miyazaki City along the coast. It is also known by the nickname "Miyazaki Bougainvillea Airport". Downtown Miyazaki is only about 30 minutes drive from Miyazaki Airport.
You may book a rental car at the Information Counter near the Domestic arrival lobby on the 1st floor. Although a booking on the day of your arrival is available, there will be no worry if you make a reservation in advance.
Since most of Miyazaki Prefecture's famous tourist attractions are located in relatively distant areas from Miyazaki Airport, it is recommended to travel via expressways. Obi is about 1 hour, Kirishima hot spring village is about 1 hour and 30 minutes and Cape Toi and the Takachiho areas are about 2 hours and 30 minutes from the airport. It is going to be a relatively long drive to go to various sights. However, you may leisurely and comfortably enjoy the views.