Before you fly off, you have to choose one of three flying heads. The Rocket Head makes you face forwards and shoots a single laser. The Air Head is a plane that lets you fire triple the amount of bullets while facing any direction. The Feather Head (which I would strongly advise against using) drops angry little birds out of your beak and, again, you always face forward.
For this scene, I would advise using the Rocket Head. I also find it useful to just keep on firing throughout the entire stage. Anyway, for the first part, you'll have Rocket Soldiers and Flying Sharks passing by. The Rocket Soldiers are easy enough to dodge. The flying sharks blast off when you fly past them, so when you see them, make them blast off to get them out of the picture (plus, you can only hit them as they blast off). When the Hovers surround the (those spheres), they'll close in on you, so when it happens, move swiftly to the left or right to dodge between them.
Next up, a mechanical arm will reach down and bring up the Sky Battleship, which is annoyingly well equipped. Firstly, there's the three guns on top, and then there's the big mouth that spews out a flying worm creature which homes in on you. If you destroy the mouth, then mini helicopters will start flying out, capable of dropping bombs that explode in 8 directions. To get rid of the battleship, you need to destroy the arm holding it in mid-air. But first, get rid of the guns on top (it will take a while). It's easier to avoid the worm creature than the bombs from the helicopters, so avoid shooting the mouth at the front. Once the guns are gone, just keep shooting the arm, while keeping an eye on the flying worm, being ready to dodge it whenever it zooms past. Once the arm explodes, the battleship will fall and you'll hear a satisfying 'splat' from below. Now go on ahead into the building for your next confrontation with Trouble Bruin.
For this scene, I'd advise changing to the Air Head so you can shoot left and right. Trouble Bruin will enter in the background on his Wheeler-Dealer. He has a particular pattern of movement that he always follows. First, when he's in the back, he'll make his way to the end of the screen, come back, and jump forward when he reaches the centre. At the front, he continues to the end of the screen, goes all the way to the other end of the screen, and then jumps back when he reaches the centre, starting the sequence again. You can coordinate with this sequence by always keeping on the opposite side of the screen to Trouble Bruin.
As for his attacks, he will fire homing missiles from the back of the screen, so just keep firing and they'll explode. At the front, he raises his head and fires a ball of lighting, so be ready to dodge it. Whenever he passes the centre of the screen, he releases bulbs on the ceiling and floor. These eventually connect by a ray of light which is lethal. But you can prevent it by destroying just one of them (the other will fall behind).
Once Trouble Bruin is defeated (hit him in the head to cause damage), he'll randomly jump to the back and start firing missiles like crazy, so keep away from him. And when you have to fly past him, watch where the missiles are coming from. Eventually, he'll stop firing, and then he'll be humorously stopped by a construction sign, allowing you to fly on ahead.
This level involves quite rapid side-scrolling, so you'll need to have complete concentration. Stay on the left side of the screen, moving up and down, and only move left and right when you really need to. You'll notice the bolts of electricity on the ceiling and floor, so avoid those. Enemies and obstacles on this level include spiders (invincible, move up and down girders), floating wheels (fly in the air once you pop the balloon holding them up), guns (some that are floating), and a type of lethal door (only opens once it's half way through the screen, so stay back).
At the end, you'll have to dodge a whole load of spiders. This can seem quite uncontrollable, but making sure you only move up and down can make it easier. Though, their may be points where you have no choice but to move back and forth a bit. Don't forget to pick up the jellybean before you leave.
Before the boss appears, make sure you have the Air Head on, and then get onto the left side of the screen, and Baby Face will appear, with the face of a baby. He'll gurgle dangerous bubbles, but if you look carefully, you should see a gap between all of them, like a pathway, so keep yourself there. Meanwhile you should be shooting constantly. It doesn't seem as if his energy is going down, but it is, very slowly. Eventually, all the bubbles should disappear, and his head will split. Stay at the top of the screen to avoid getting hit by the face flying off.
This new head will move to the right side of the screen, and as soon as he is there, a finger will burst out from below you, so be ready to move out of the way. This will then descend while shooting wiggly lines, and the head will move to the opposite side, as should you once it has bobbed up and down. Again, the same thing will happen, only, on the opposite side of the screen. You'll know when this head is beaten because he stops moving to the other side of the screen, so go to the top of the screen again, to avoid the old head hitting you as it flies off.
This next head moves to his permanent place on the right side of the screen, and starts firing lasers at you (one line of which will shoot multiple lines), so stay away from him so you have a better chance of dodging them. To cause distraction, a hand will come up in the middle and begin waving around a sign saying something like 'Headdy the Hero', but don't be tricked into reading it and losing focus. Instead, just keep an eye on where the lasers are going, and where you should be going to avoid them, and you'll eventually be able to move onto the final head.
This head, is probably the easiest, because he'll send a withered old hand up to grab you, But you can dodge this by going to one side of the screen, firing at him, and moving to the other side of the screen when you see a disturbance in the clouds below. If he does catch you though, he ends up dying anyway. Defeating him yourself will simply give you the opportunity to get a Secret Bonus Point by shooting Bino, who is tied to the pole inside the final head.