Go right as soon as you enter the level. Pick up the DK Barrel if you need to, then continue right to trigger a cutscene.
When the fight starts, the third world boss, Stu, will fly around and drop bombs. You can throw these bombs at Stu to cause damage, but be careful when they start flashing orange, as they are about to explode. The best places to throw bombs from are the second and fifth platforms, as they give you just the right amount of altitude and distance from Stu to make contact.
After dropping a few bombs, Stu will dive down to attack. Dodge him by hiding between the first and second (or fifth and sixth) platforms or by ducking. This is also a good time to throw bombs at him if you're carrying any. Stu can also drop bombs that have a flame on them. These bombs explode faster than normal, and when they explode, they create a flame that moves along the ground and climbs over blocks. Jump over the flame and it will disappear when it hits the left or right walls.
Once Stu takes three hits, he will drop a giant bomb in the middle of the field that destroys the six gray blocks. After that, Stu has a couple of new attacks. He can drop a slow-moving Tiki bomb that you can't use against him (just dodge it), and he can drop an entire row of bombs and dive down immediately after. When this happens, jump between two golden blocks and throw out any bombs inside. If there's enough time before the bombs explode, you can throw them at Stu when he dives down.
After hitting Stu six times, you'll win the fight. Jump up to the Tiki and knock it out to clear the level and progress to World 4.