Since the rising action is what leads up to the climax (or the moment of most tension) of the story, in order to ascertain what the rising action is, it might be helpful to first determine where the climax occurs.
I would argue that the moment of most tension in the story is when Montresor finally reveals to Fortunato that he has been tricked as a result of his own pride and will soon die because of it. Montresor chains him to the wall, and Fortunato is still too dazed to understand at first. Montresor says to him,
"Pass your hand [...] over the wall; you cannot help feeling the nitre. Indeed it is very damp. Once more let me implore you to return. No? Then I must positively leave you. But I must first render you all the little attentions in my power."
We can imagine him saying this with a terrible sneer since Fortunato obviously cannot now turn back. All along, Montresor has been -- seemingly earnest -- imploring Fortunato to turn back knowing full well that he will not. Montresor pressed him to consider his health and the dampness of the catacombs, and he's even suggested that they should "'go back ere it is too late,'" some major foreshadowing of what's too come. He knew, however, that Fortunato's incredible pride would keep him going until they reached the Amontillado, and it is at the moment cited above that Fortunato begins to understand what Montresor has done: exploit his one "weak point," his immense pride, to get him to walk right into his own coffin.
Therefore, all of the action leading up to this moment would be considered the rising action: Montresor affecting to bump into Fortunato on the street, taking him back to his home (which he ensured would be empty of servants that night), walking through the catacombs, getting more and more drunk, all of the times Montresor suggested that they turn back, and so on up until the moment when Montresor reveals the purpose of all of these actions.