How do you decide the minimum annular ring size of a buried via?
I remember to have seen one manufacture list this, but generally I can't find the size on manufacture websites.
Generally, the annular ring spec, for buried vias or through vias, is driven by the accuracy of the drill hole locations relative to the copper features. You want to have a ring big enough that even with the maximum error in the drilling location, the drill hole is still completely surrounded by the pad (some rules might allow the hole to "break out" of the pad by a small fraction of it's circumference).
So if your buried vias are drilled on the same machine as your through vias, I'd expect to have the same annular ring requirements. If for some reason your shop uses a different tool for drilling buried vias, they may give you a different requirement. Also, of course if your buried vias are laser-drilled or something, of course there could be an entirely different spec.
So, like most questions on PCB tolerances, it comes down to asking your shop. Even better, ask several shops and use a rule that will give you flexibility to use whichever one gives you the best bid when it comes time to order the boards.
Best thing would be to ask your PCB manufacturer about their capabilities.
Also, don't just go with the smallest via they can make, use the biggest one you can put in your design within those tolerances. That way your design will be better for DFM.