I think he just told the story as he lived it, but it might not be what had happened. From the very beginning of the story, he tries to make a case for his sanity, but the story he tells completely undermines it and is at odds with his declarations of sanity. Throughout the story, he recalls the events that led him to murder the old man and then confess. According to the text it states, "'Villains!' I shrieked, 'dissemble no more! I admit the deed!— tear up the planks!—here, here!—it is the beating of his hideous heart!'"
I'm not sure what text this refers to but in a general context, yes the first-person narrator is trustworthy when he gives his account of the story if s/he was at the place s/he are present at the event they are talking about/narrating
I'm sorry if this isn't much help but its hard to give evidence without knowing what text you are referring too. Hope this helps. Let me know if you have any questions
