Can I travel by plane with my residence permit, ID card, Geprivelgeerde Dutch card from Holland to Budapest, with my expired passport?

I don't think you are entitled to travel with just that. For example, the official guidance for law enforcement personnel explains that

De geprivilegieerdenkaart van Buitenlandse Zaken geeft, samen met een geldig reisdocument, de houder het recht het Schengengebied binnen te komen (en daar te reizen).

In English:

The geprivilegieerdenkaart from Foreign Affairs, together with a valid travel document, grants the holder the right to enter the Schengen area (and to travel there).

This means that the geprivilegieerdenkaart basically exempts you from any visa requirement. While it does not explicitly say anything about internal Schengen borders or unambiguously state it's not valid without a passport, it's strongly implied that one is needed. And of course this only covers the Dutch side of things.

Without a valid passport, a European (EU, EEA or Swiss) national ID card would also be enough but a South African ID will most likely not be recognised.

But on the other hand, you should not have to go through any official passport check. If you would travel by road, there would still be a slight risk of encountering some sort of random check but generally speaking you should be able to reach Hungary without problems.

By air, the airline/ground handling personnel will probably want to see some ID but they might not be aware of all the details of the rules and the geprivilegieerdenkaart is in fact a valid form of ID in the Netherlands (although not necessarily in Hungary). So it might just work. But you can't count on it.

The Dutch national ID card, can indeed be used as a travel document within the EEA and Switzerland. This is true for various other ID cards released by countries inside the EEA and Switzerland. Since the EEA includes the EU, and Hungary is inside the EU you should be able to fly to Budapest with your Dutch ID card.