Dodge the first giant mace, then there are the two other ones that are much harder to avoid.
If you don't already have a spare power-up, pick up the mushroom to the right of the door to Part 2.
Part 2
As soon as you get through the door, run. There's a spiked ceiling rushing down. As soon as the ceiling is just above the ON switch, turn it OFF. Dodge the Thwomps and fireballs.
The ceiling will come down again, and the path between the lava pit and the safety zone is huge! Dash extra quickly!
Part 3
This room is similar to Part 1 of Iggy's Castle, but this time you advance vertically.
Hit the Koopas on the fences. Move to the right for a "?" block with a cape (or a mushroom if you're small).
You will need to jump from one fence to another this time. However, you can still score a 1-Up by disrupting fence Koopas. Jump some more fences until you get to the final door.
Part 4 (Ludwig's Room)
This room is two screens wide, so at first you won't see Ludwig.
Dodge his fire balls and jump on him. He'll retreat into his shell and jump.
If you're short on time, or if you don't like to waste time, make sure you Ludwig doesn't go halfway offscreen. After he jumps, repeat the process 3 times until he is defeated.