The old man, mentioned in Parts II and V, is showing the unnamed first person narrator around the property; he has keys to the rooms and intimate knowledge of both the layout and lore of the villa. Though his actual connection to the house is not mentioned, one may conjecture that he is a grounds keeper or caretaker of sorts. The nested narrative, and so the greater part of the story, comes to the reader via the old man telling the narrator the events that took place there roughly 200 years prior.
The unnamed first person narrator describes the old man thus: “He was the oldest man I had ever seen; so sucked back into the past that he seemed more like a memory than a living being. The one trait linking him with the actual was the fixity with which his small saurian eye held the pocket that, as I entered, had yielded a lira to the gate keeper’s child.”[1]
He seems to be a devout or perhaps superstitious man as he crosses himself when he and the narrator survey the statue's contorted face.[2]
He hints at the Duchess's story but seems to only want to tell it to willing ears, as seen when he appears to calm up after he mistakes the narrator's "bewilderment for incredulity". He only divulges the tale after the unnamed first person narrator jingles the lire in his pocket (implying that he will get something out of telling the tale)[3].
From these instances, it seems that the old man is a little greedy and, though he feigns being put off at first, eager enough to recount the events that transpired at the villa.
[1] Wharton 1901, p. 2
[2] Wharton 1901, p. 4
[3] Wharton 1901, p. 5