How do you deal with the costs of too-rapid change?

Question

Like most modern developers I value Agile principals like customer collaboration and responding to change, but what happens when a product-owner (or whoever determines requirements and priorities) changes requirements and priorities too often? Like several times a day?

I recently inherited a smallish code base that was buggy, incomplete, and couldn't even handle the simplest scenario it was supposed to. I can deal with the technical issues but I get several emails, texts, or phone calls a day saying "OMG you MUST work on this RIGHT NOW! TOP PRIORITY! This is a MUST!!!oneone" (that's only a slight exaggeration) What makes it even worse is that most of the things are minor details that aren't even relevant to what the software is actually supposed to do and would take days to implement anyway. I've tried explaining that there's only so much time and that we should focus on the most important things first, but something seems to get lost in translation because the same thing happens a day or two later.

Is there some sort of Product-Owner-Handler role, in-depth study, metaphor, or quote that can help me reduce the amount of wasted effort or at least explain the costs of this chaotic behavior?