There are two easy ways to get through this game and this is the easier of the two (IMHO):

Play the game through 1-3. In 1-3, there is a large building at the end of the outside area. To the left of that building is a small island. There is a potion on that island. Do not throw it yet. Walk to the left, all the way to the end of the building. You will see a pot (the kind you can go into). Throw the potion next to the pot. In the sub world, enter the pot. This will take you to world 4-1.

From 4-1, play all the way until 5-3. Select Luigi to play for this level. Start the level and exit the first door. Above the outside door, there is a pot. Jump up to that level, and look for a potion nearby. Throw the potion near the pot, enter the door, and then jump in the pot. This takes you to level 7-1.

From here, finish the game as you normally would.