Practice maneuvering the Explorer to become familiar with the way he moves and how he jumps. You'll need to be both fast and agile when you're trying to escape with the Crown.
Listen carefully to the sounds and the music. They let you know when the Spider is approaching, how close you are to the Flame Spirit and how much time you have left to reach the mountaintop.
Keep track of where Treasure Chests are located and wait until later in the game to open them up. You'll soon discover your dependence on them as you begin to deplete the mountain's supply of diamonds.