Use http://reiseauskunft.bahn.de/bin/query.exe/en  the German rail website for schedules anywhere in Europe.  Sometimes it’s a bit tough to work out the correct spelling for the town as the German’s tend to use the Russian, as opposed to Ukrainian, spelling and then transliterates that.  From the Hungarian border is going to be a long trip.  I’d look at flying — perhaps Austrian Air, via Vienna.  I’m not sure where you can find pricing for tickets in Ukraine, but when we were there in 2004, trains were dirt cheap.  We spent about $60 US for the overnight train from Kharkiv to Sevestapol.  As I recall, that was for 6 of us to buy out 2 4-berth compartments, so it must have been less than $50 for each berth.   The trains in Ukraine are Russian gauge, so the cars are much larger and much higher than the standard European train — that might cause a problem at the Hungarian/Ukraine border as they either have to switch the wheels on the train or you have to switch to a different train.
 
Some of the trains in Ukraine are the old straight out of Dr. Zhivago Russian green or Ukrainian blue cars.  They are old, but well maintained, clean and comfortable.  Other trains are very sleek and new — on a par with the fanciest trains in Western Europe.  Trains run slower that the high speed trains in Western Europe, but they’re faster and much more comfortable than the bus.  The roads are, for the most part, truly horrible.  Riding in a bus or car is a slow and bone-jarring experience. Buses don’t necessarily run on a set schedule.   Hopefully, roads have improved some in the last 3 years, but the train is definitely a better option than the bus.

Probably a full day.  But it would help if you were a little more specific than some Hungarian town.  Lviv to Kiev is about 12 hours I think.  Lviv-Budapest was supposed to take about 8 or 10 hours, but I missed a train and it ended up taking about 20. 
 
I think there is only one train crossing between the two countries, Zahony-Chop.  Ukraine is still dirt cheap on train fares, Hungry is reasonably priced. 
You could save a bit of money by buying one ticket to the border and then another, but its a little more difficult that way because no one in Chop will likely speak any english and they will only accept Ukrainian currency.  There were two different train station buildings at Chop, the first one has customs check, a short walk down the rail line is the other one and I don’t know what its for but it has an ATM in it (and better heaters!).