If you plan to travel by plane to Hungary choose Budapest Ferihegy Airport as your destination, and then you can easily reach Eger by train.
Airport Minibus service can get you to Budapest-Keleti railway station, from where you can pick up a train. To search rail service, you can use hungarian railway service searcher
GPS coordinates: Lenght: 20o22' 56,66" Latitude: 47o53' 54,81"