Regardless of the time of day you decide to fly, the ticket price will remain around the same. Book flights from Qatar to Košice as you normally would.
For a flight from Qatar to Košice, you will be flying into Barca. Košice Barca (KSC) is just 5.8 km from the centre of the city.
This is defined as Doha Hamad Intl Airport (DOH) and Košice Barca Airport (KSC). Codes like Košice Barca Airport are often used to just call airports uniquely.