It says in paragraph 2, "I made up my mind to take the life of the old man." We can prove he is sane because he had decided to kill the old man. He knew right from wrong, which proves he is sane.
"Villains!" He shrieked, "disassemble no more! I admit the deed!- - tear up the planks! Here, here! - - it is the beating of his hideous heart!" Paragraph 18 shows that he was guilty, and if your insane then you wouldn't feel bad if you didn't know it's wrong.
In paragraph 6, "for a whole hour I did not move a muscle, and in the mean time I didn't hear him lie down." This proves that he is sane because he knew that if he moved, he would scare the old man.