You can buy maps in bookstores. I am pretty sure there is an Alexandra or Libri bookstore in Eger. There you can find a larger selection of maps, specifically about walks. In Hungary there are colour-coded walks (blue, green) that are going right across the country touching important and significant places. It goes through the Bukk National Park. You can pick any part of these walks and gives you information about elevation, roads the whole lot. I personally prefer the fells around Miskolc. Lillafured, Omasa, Bukkszentkereszt, the Hamori-lake and the waterfall at Lillafured are really great. The fells are excellent around there too.

Dear Shoshi, There are several designated walking trails of varying lengths and difficulties, all of these designated trails are equipped with fixed maps but i am sure that paper maps would be available at the numerous souvenir shops, cafes and watering holes throughout the park. Alternatively you could take the train, which covers three different routes, outbound and walk back. We on the other hand were more spontaneous and just went where the trail took us. There is just so many beautiful thing to see, these change with the seasons but make a point of seeing the chateaux {now a hotel} and the beautiful gardens, the lake and the waterfalls. There is also a wonderful restaurant attached to the hotel with great food and very reasonable prices. Hope this helps you and have a wonderful trip.

The most popular is túristatérkép Bükk from Cartographia. You can buy it in any bookshop, not just in Budapest.