How is a biblical papyrus manuscript dated?

It is better to say something else. Papyrus was used for a defined period, roughly a hundred years before Christ to about a hundred or two hundred years after, so if a gospel text is written on papyrus, we know it belongs to that early window. Vellum, the treated leather used for the great codices like Vaticanus, Sinaiticus, and Alexandrinus, was not used until the fourth century and lasts much longer, because it holds more moisture and does not get brittle, whereas papyrus dries out and becomes brittle. So dating a papyrus specimen depends on factors like how brittle it is, the author who claims to have written it, where and how it was stored, and how much it deteriorated, including from the sweat of the hand as it was written. Because of that early window, almost any New Testament text written on papyrus can be taken as the oldest copy of that passage, since they had no vellum yet, only papyrus. Interestingly, a few papyri are actually older than some vellum texts, which reminds you not to assume too rigidly.