Your best bet is to go to the Puławy PKS and look around for the international connections. There might be a posted schedule outside somewhere as well as offices inside. It's hard to do on line but much easier in person. The address is Lubelska 9, B on this map.
I doubt there is a direct bus connection Puławy-Budapest
I couldn't find any reliable source on bus connections from Poland to Budapest
but I found a direct night train from Kraków to Budapest - I guess you can book and pay for the journey in Puławy railway station - the train
rozklad-pkp.pl/traininfo.php/pn/93414/261087/355434/146579/4095 /?q=pl/node/146&ld=pkp&seqnr=4&dent=6b.01420529.1314180910&(start s just before 10 pm in Kraków and arrives after 8 a.m in Budapest - seems like it is a daily train) - you would have to take someone who speaks Polish with you to make a reservation and buy the ticket for the train - you need to make a reservation for the train and I guess the sooner you make it the better
you can reach Kraków from Puławy either by train or by bus - learn what suits you better - in case of train journey from Puławy it perhaps would be possible to buy a ticket for the whole journey up to Budapest - should save you money a bit
gumishu
busses to Budapest seem to go only once in a week - here you can find some connections (I suggest Kraków for a departure point) you can also presumably book online here (prices are from 80 to 200 PLN) biletyautobusowe.net.pl
Yeah, actually the night train to Budapest (from Warsaw as well) works fine. I've taken it a number of times (both directions). The conductors of the sleeping (or couchette) car are IME Polish about 2/3 of the time and Hungarian 1/3 of the time. But if you have your tickets language isn't much of an issue. Typically trains from Poland arrive at Keleti palyaudvar in the morning.
