Since he is the one figuring out the criminal, is the narrator jealous of Dupin? Is the narrator friends with Dupin?
Based on the information that's provided in the story, I don't think that we can justifiably state that the narrator is jealous of Dupin. We might, in the narrator's position, feel that way, as Dupin certainly has all the elements and attitudes of an enviable person (and some arrogance too) but the narrator does not express this at any point.
The narrator does express, in the extended introduction, a sort of introspective social hypothesis regarding the different mental acuities and dispositions of people in general; he feels that this theory is evidenced by his interactions with Dupin. Note that at no time in his theorizing does he presume one type of mentality is superior.
The narrator is certainly friends with Dupin (they live together amicably) and he frequently remarks on Dupin's abilities with no suggestion that he is jealous (envious, perhaps). In fact the narrator might almost seem to be fawning and fan-like, since he is a thin character on his own, and his expository statements seem intended only to magnify Dupin's character.