The only thing I can think of - the idea of replacing one holder of a role with another? Although I realise that it's not the same tone as the French phrase, with the reference to Iznogoud.

Hello
I liked the "King is Dead" attempt but can it be made to work? I'm not sure as the bloke was promoted.
She gets his job but she is not keen so for me it's like the monkeys climbing up the ladder.
Other monkeys are still higher up to shit on you and you are still clambering, albeit just a bit higher up

I think this conveys the ironic tone of the newly promoted person, with a bit of humor.

He is promoted so I move into his/her position.
Caliph = The civil and religious leader of a Muslim state considered to be a representative of Allah on earth.

This isn't exactly what you're looking for, but here goes anyway. The irony in the FR expression is very much a cultural insider joke if the speaker isn't in fact keen on her supposed promotion—like Iznogoud finding out that being Caliph isn't so great after all.

It sounds like she'll still be fairly low in the food chain, hence her lack of enthusiasm in part, I guess?

"Big news" is meant to convey some of the irony, and if possible or appropriate you could perhaps add something like "Yay" or "Big whoop" afterwards, but at the risk of losing subtlety.

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Note added at 16 hrs (2020-06-05 11:15:02 GMT)
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Don't mean to split my answer in two, but perhaps "greasy pole" might also work in your situation:
" [...] he gets a move-up, I get his job, that's the greasy pole for you."
" [...] he got a promotion, I got his old job, so much for the greasy pole."

Hard to tell if the speaker is half-hearted about her new title or the job itself, but I imagine the point is to convey her feeling that moving up in the world isn't all it's cracked up to be, whether in general or in her own case.