The narrator's bravery is a different kind of heroic bravery. Normally when readers think of bravery, they think of super heroes facing certain death and smashing a bunch of stuff in order to get the bad guy. Usually, that kind of hero is very emotional and is trying to protect other people. That's not the case with Poe's story. The narrator is only trying to protect himself, and he never once has to resort to violence and smashing things.
The narrator's bravery is displayed through his unwillingness to give up and in his refusal to panic. Throughout the story, the narrator brings to bear a cool and calculating logic.
I still lay quietly, and made effort to exercise my reason.
He knows that his situation is bad, but he never panics. He might be scared, but he doesn't panic to the point where he can't think. Each time the situation changes, he thinks his way toward the solution. He is able to do this by remaining calm in terrifying situations. That's bravery. And let's be honest, he goes through some fairly horrific stuff. Most notably is the pendulum swinging its way down toward death all while he is swarmed with rats.
They pressed -- they swarmed upon me in ever accumulating heaps. They writhed upon my throat; their cold lips sought my own; I was half stifled by their thronging pressure; disgust, for which the world has no name, swelled my bosom, and chilled, with a heavy clamminess, my heart.