One minor complication, is "the oldest man in the world" a name or a description? It reads to me like a description. In that case the question becomes whether descriptions can be rigid designators.
Kripke, in arguing against the descriptive theory of names in Naming and Necessity held that descriptions couldn't rigidly designate. That was one of his objections to descriptivism. If all names are hidden descriptions, and descriptions aren't rigid (i.e., don't refer to the same individual at all possible worlds) then we will have to give up on the necessity of identity, for instance. Even though Aristotle was the teacher of Alexander the Great, someone else could have taught him and so there is a world where "the teacher of Alexander the Great" doesn't refer to Aristotle.
On the other hand, you could hold that there is a way of rigidifying descriptions, or that certain kinds of descriptions are rigid. One thing I've always thought a bit odd given Kripke's views on the essentiality of origins: why wouldn't "the person born to such and such on such and such a date" be a rigid description? If descriptions can be rigid, then you can use the description to refer unambiguously to the same individual, without the reference switching to the next oldest man upon his death.
Although it seems to be (or at least was around the time of Naming and Necessity, I haven't followed the contemporary literature here) a matter of controversy among philosophers as to whether descriptions can rigidly designate, many linguists I've talked to take it as a given. I'm not quite sure the import of this, but I've been told that there is plenty of linguistic evidence that we do use descriptions rigidly (like in your newspaper headline).
TL;DR If the description is rigid, then the oldest man in the world can die. If the description is not rigid, then the referent of the description will change once the current oldest man in the world dies. So, in a sense, although the person who once satisfied the description will have died, "the oldest man in the world" cannot die.
"Description and Identification": in this article Bernard Harrison claims that the descriptivist theory of names can accommodate rigidity, precisely because some descriptions are rigid designators.
This paper is a response to Harrison and argues that the descriptions Harrison claims are rigid, are in fact not.
In sum, I see this not so much as a question of descriptive vs. causal theories of names, but the (admittedly very closely related) question of the rigidity of descriptions.