The gender of the unnamed first person narrator can safely be assumed to be male as he is referred to as 'he'. His role is mainly that of an inquisitor. His questions prompt the nested narrative from the old man in Part II and their conversation about the events at the villa continue in Part V.
He seems reasonably generous, having given a lira to the gate-keeper's child[1], but also make the impression of being a little unkind. His descriptions of the old man feature words with more negative connotations and he notes the old man fixing the pocket the lira was withdrawn from with his 'small saurian eye' - evidently his generosity does not extend to all.
He is being shown around the villa at Vicenza by the old man. One can only guess as to why; evidently the villa is still in a Duke's possession[2] so it seems unlikely that he is there to purchase it. As he seems to have lira enough to frivolously gift to children, it also seems unlikely that he is there being interviewed as a type of servant or house keeper. Since the especially the Duchess's apartments, and later the statue, are so reverently and carefully revealed to him by the old man, it also seems unlikely that the unnamed first person narrator is merely a tourist or no import - if he were, why would the old man be giving him a private tour of villa and divulging the origins of it strange lore?
He seems to be an inquisitive fellow. He asks the old man many questions and asks for clarification in multiple instances. Evidently he is quite interested in the house and the stories it holds within. The readers of course profits from this as it is the first person narrator's curiosity that inspires the old man to relate the tale of the Duchess.
[1] Wharton 1901, p. 2
[2] see Wharton 1901, p. 2