There are five possibilities that I can think of. You can hire a driver from Budapest. The one I know and like a lot is named Laszlo Markus. http://silverwings.hu/ You could drive up and back in a day if you wanted. He’s great but it’s a bit pricy. I am guessing $400+
You could hire a guide and see some of the great stops in the area. You will be Eger and Tokaji and Lillafüred. Andrew (Andras) ILLES is my favorite guide in town, but there are a number of good ones. http://www.guideinbudapest.com/prices.html
With either of those options you could do a day trip or spend the night in Eger or if you are into nature maybe Lillafüred (stay at the Palota Hotel https://goo.gl/images/odM2ts ) and visit the caves and ride the forest train.
More creative, but less exploratory would be to take a train to Kosice, Slovakia http://elvira.mav-start.hu/ and find a guide there. There is only one direct morning train at 6:30 and it takes 3:30 and I suspect it will cost about $20. From Kosice, Göncruszka is under an hour drive. Hana is the only guide I know in Slovakia: http://vasvarirentals.com/img/partners/slovakia.pdf but I have no idea what she would charge. If you do this spend the night in Kosice. I understand its interesting and worth the night.
There is a train to Göncruszka. One train only takes 4 hours and has two changes, the rest are pretty long rides. But what do you do when you get there.
Finally you could take the train to Eger and hire a driver or a guide from there. The direct train takes about 2 hours. I dont know any Eger guides but its a tourist center so i bet you can find one. Nice place to overnight too. If you choose this way and cant fine a guide let me know and i will ask around. You could use a taxi I guess. As for is the taxi safe? Absolutely. But i bet sort of expensive, but i could be wrong. City Taxi has an operation in Eger. City Taxi is my most trusted and relied upon taxi company in Budapest and if their Eger office is as good then you wont do wrong. http://www.citytaxieger.hu/ I think you would get more out of it with a really good English speaking guide or driver.