I have spent a lot of time with very rich people over the years for work and they will certainly bite it incredibly early in the apocalypse. If someone is born into wealth, they are almost always fundamentally inept in basic ways. The longer someone stays rich, the more inept and abstracted from the world they become. They are entirely ensconced in a bubble of their own making. Money opens so many doors and protects them from so many just basic consequences of living that they entirely forget that those obstacles exist.

They are also just constantly, constantly being glad-handed and yessir'ed into an excessive amount of confidence. They know it too, but they can't really trust anyone so they just make friends with people who they know would steal the wallet off their corpse but happen to be nicer about it than the other ones who would also do it because they have nowhere else to turn. Other rich people aren't trustworthy to them either because they are only either more or less rich, above or below them.

It'll go down pretty much how it did in the last third of the movie Triangle of Sadness. They are building bunkers that their handlers and others will actually live in. Not to mention that anyone that knows they were a potential engineer, actively or passively, of the downfall of man will have someone to take it out on right in front of them.