score:4
I recommend you have a look at rome2rio which find both bus and train connections between those two cities.
Trains seem to go every two hours, take 7h of total traveling time and cost €26, changing twice in Püspökladany and Biharkeresztes.
Buses are direct either from Budapest Nepliget bus station (easily reachable by metro from anywhere in Budapest) or from Budapest airport (reachable by bus+metro from Budapest city) taking about 7h and going four or five times daily. Bus operator in both cases is the Romanian presidenttour.ro and cost seems to be around €20.
Upvote:2
You can fly to Targu Mures with Wizzair (can be as low as 5990 HUF around 20 EUR, I see that for sep 30 https://wizzair.com/en-GB/TimeTable/Budapest?arrivalIATA=TGM) and take the bus (25 lei around 5.7 EUR) from Targu Mures to Cluj. Beats sitting on a bus for seven hours.