I actually know where Mukacheve is located and it’s not too terribly far from the Hungarian border. Not that I have ever been there, but I have been working on plans for a fishing trip to Ukraine in the spring. I have my set of limits when I travel and the hassle of taking the train to Ukraine exceeds those limits. Not impossible or absolutely horrible but not my cup of tea. Although the train from Budapest does cross about an hour away in Chop, Ukraine. I believe that's where they change the carriages to the Ukraine narrow track system. Then from Chop it appears as though there is a track to Mukacheve. I bet you could make the trip from Budapest in under 30 hours by train.
The town of Mukacheve is about an hour and a half from Nyíregyháza, Hungary. You can reach Nyíregyháza in about 4 hours by train from Budapest. From there I would hire a Ukraine guide/driver. But actually, no, I wouldn’t do any of that. We will be flying into Lviv from Budapest (changes in Vienna – about 4 hours). If you are going to Ukraine Lviv is sort of a mandatory stop as the city is reportedly very worth the time. From Lviv to Mukacheve is about 3.5 hours by car through what appears to be some pretty amazing country. We will hire a driver for this ($200). My fishing location is on the same route but just a little closer to Lviv. After the fishing we are returning to Lviv, then flying to Kiev to look around and show support, then flying back to Budapest (changes in Munich – about 4.5 hours). Total airfare is about $375.00.
Just because grandma was from Ukraine doesn’t mean she isn’t Hungarian. There are ethnic Hungarians in Ukraine; our you might have a little Hutsul blood in your veins.