Okay... I think you are actually playing the original Legend of Zelda; as a Link to the Past is the SNES game and you are mentioning stuff that sounds like the first game...

I believe the Magic Sword is acquired either when you get a certain amount of hearts or completed 6 or 7 of the dungeons. I think the Old Man in the grave is the one who gives you the master sword. I can't remember the third dungeon. I remember where all the others are except the third dungeon....