Skoll Shorties

Since anubarak is kinda a descendant of oldgod spawns I doubt he'd have a "soul" that would go to the shadowlands

Wouldn't that mean Vash'j wouldn't go there too? Plus the nerubians all renounced the Old Gods so wouldn't that mean that they'd be in the clear for going to the Shadowlands even if they were formed or created by them?

Not really, Vashj was a highbourne, she just made a nasty deal. Old gods come from the void lords, and the nerubians from old golds, I dont think it's a matter of who you're alligned to

However there are Naaru in the Shadowlands as well, alien beings not native to Azeroth. And one of the Maldraxxus NPCs mockingly talks about how the Void (Old Gods) and the Light (Naaru) do not really matter there, so its possible Anub'arak would end up somewhere there.

I dont think they're there because they died. I think they can access it due to their cosmic nature

Perhaps, but the one in the Sanguine Depths definitely is said to be a prisoner of the Denathrius Venthyr, which seems surprising if it wasn't sent there after dying. Naaru can physically die if their forms are destroyed, like the one Velen's son kills.

Then again another theory I like is that the Naaru have their own realm in the Shadowlands and they also try to collect anima/souls to fuel their own agenda.