Now that the Sarajevo-Budapest train has stopped running (and given that there's no direct bus) what is the best way to go between Sarajevo and Hungary? Someone suggested to start with a bus from Sarajevo to Novi Sad but in the main bus station I could see no such bus (maybe there's one from Lukovica/Istocno Sarajevo bus station?) Any other ideas?
If you have plenty of time, you can take the night bus operated by Croatia Bus, departing 22:00 in Sarajevo and arriving 06:00 in Zagreb. Then the train from Zagreb Glavni Kolod. at 09:57, arriving in Budapest 16:29. The price will be about 55€ (25€ for the bus and 30€ for the train).
This answer comes super late, but these are my suggestions. Take a bus to Belgrade, then either a bus or a train to Budapest from there (around 20-25 euro). To get you to Belgrade, take a bus from Istočno Sarajevo (around 25-30 euro return ticket). Novi Sad only has a train to Budapest (I think I paid 20 euro for the return ticket). Another, more expensive option, is taking a flight from Sarajevo directly to Budapest. Without luggage it's about 50 euro (you may have one carry-on free, you should check on the Wizzair website). With luggage it's about 100 (ish) euros, maybe even cheaper. There's one or two flights per week.