Wizz Air flies SJJ-BUD these days Monday and Friday.
Car share is a popular optinon. https://www.motar.eu/ and https://www.blablacar.com/
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.
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).
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