Transportation

For your help here are the ways of getting to Eger from Budapest:

Hungary is connected to all of its seven neighbouring countries by road and rail. Volánbusz (Coach Company) runs regular transport services to about 18 different European countries, while Magyar Államvasutak (MÁV, Hungarian Railway) has express rail services that run as far as London, Stockholm, St Petersburg, Istanbul, and Rome.

By Air

Malév Hungarian Airlines, the national carrier, has direct connections with more than 30 European countries.
Information about timetable: www.bud.hu/english
The main international airport in Budapest is the Ferihegy Airport, about 20 km from the centre of the capital. Taking a taxi into Budapest from the airport might end up costing you a lot more than it should, but Airport Minibus is a reliable shuttle express service that runs to and from the airport. It is relatively inexpensive, runs frequently, and you can find out the schedule at any airport information desk.
Information about travelling within Budapest: www.bkv.hu/en/

By Rail

Trains leave to Eger at Keleti Pályaudvar (Eastern Railway Station – at the Keleti Pályaudvar stop of the red metro line). MÁV provides normal and Inter City services to Eger according to the timetable. The other two railway stations of Budapest: Nyugati Pályaudvar (Western Railway Station – at the Nyugati Pályaudvar stop of the blue metro line) and Déli Pályaudvar (Southern Railway Station – at the Déli Pályaudvar stop of the red metro line).
Information about timetable (inland travels): www.mav-start.hu/english/index.php

By Coach

Coaches leave to Eger at Stadionok Buszpályaudvar (Stadions Coach Terminal – at the Stadionok stop of the red metro line) according to the timetable. There are buses to Eger almost every hour between 6 AM and 9 PM.
Information about timetable: http://www.menetrendek.hu/cgi-bin/menetrend/html.cgi