Yes, Montresor achieves exactly the kind of revenge he wants. He explains what he wants in the opening paragraph of the story, and by the end of the story he appears to be fully satisfied with what he has done. In the opening paragraph he states:
At length I would be avenged; this was a point definitely settled--but the very definiteness with which it was resolved precluded the idea of risk. I must not only punish but punish with impunity. A wrong is unredressed when retribution overtakes its redresser. It is equally unredressed when the avenger fails to make himself felt as such to him who has done the wrong.
In more simple language, he wants to kill Fortunato without getting caught. The word "impunity" suggests that he wants to feel completely safe not only from exposure but even from the slightest suspicion of being guilty of the crime. He also wants his victim to be sure that it is he, Montresor, who is responsible for his victim's murder. In other words, he doesn't want to do something like sending Fortunato a bottle of poisoned wine or hiring some assassin to kill Fortunato in a dark alley. Montresor wants to do the deed himself with impunity and have Fortunato--but only Fortunato--aware that he is killing him for revenge.
Montresor has a lot of problems achieving his perfect revenge, and his coping with all his problems is the essence of the story. He lures Fortunato down into the catacombs and chains him to the granite wall. He manages to do this without being recognized by anyone in the streets above. Fortunato was drunk when Montresor encountered him, and Montresor keeps him drunk until he has him in chains. Then Fortunato realizes he is in deep trouble and quickly sobers up. This was necessary for Montresor to be sure that Fortunato knows what is happening, why it is happening, and who is doing it.
It is noteworthy that Fortunato never calls Montresor by name until he is chained to the granite wall. Then he cries:
"For the love of God, Montresor!”
“Yes,” I said, “for the love of God!”
Montresor says "Yes," because he now has proof that Fortunato recognizes him as his killer. Montresor adds, "...for the love of God!" because he is pleased to have his arrogant victim begging for mercy.
At the very end of the tale Montresor states:
Against the new masonry I re-erected the old rampart of bones. For the half of a century no mortal has disturbed them. In pace requiescat!
The fact that no one has discovered the bones in fifty years proves that no one has ever discovered the wall or ever suspected what is behind it. Montresor has achieved his revenge with "impunity." The purpose of getting revenge was to rid himself of all the painful thoughts and feelings that made him want the revenge in the first place. Now that he is fully satisfied, he means it sincerely when he says, "In pace requiescat!" (Rest in peace.) The Latin words are meant to convey the idea to the reader that Montresor did achieve the kind of revenge he wanted and feels utterly cleansed of his hatred and rage.