The First Mutant: Apocalypse's powers and combat abilities can't be countered, and he can't be the target of Incapacitate. Apocalypse can use Plasticity and Shape Change.

You Appear More Worthy: Once per game, give Apocalypse a power action and choose two characters: an adjacent opposing character not on its starting line and a friendly character on the map with a point value higher than the opposing character but lower than Apocalypse. The first character becomes friendly to Apocalypse and gains the Horsemen of Apocalypse keyword. The second character becomes friendly to the first character's force and is healed of 1 damage.

Survival of the Fittest: Apocalypse can use Mastermind and Invulnerability.

Centuries of Scheming: Apocalypse can use Outwit. Apocalypse deals penetrating damage to any characters who have a power countered.

See the Colossal Stamina and Great Size entries for more information.

What exactly does You Appear More Worthy do?
If there is an opposing character adjacent to Apocalypse that is NOT on its first click and you have another character on your team that is more expensive than that adjacent opposing character, Apocalypse can make that adjacent character part of your team and give your opponent the more expensive character.

In a nutshell, if there is an opposing character that you would like to have on your team (maybe it is better suited to fight the rest of your opponent's team, maybe it is just being a pain to deal with, whatever), you can swap one of your own characters out to get that character.

Can Apocalypse use Invulnerability to reduce damage before he transfers it with Mastermind?
No. Mastermind activates and transfers damage when damage would be dealt (AKA, before it is actually dealt to Apocalypse). So if you use Mastermind, the damage is transferred before Invulnerability would have a chance to reduce anything.

Does Apocalypse only deal penetrating damage to characters he has countered? Or can he deal penetrating damage to any character with a countered power?
Any character. Nothing in the description of Centuries of Scheming says that Apocalypse must be the "source" of the countering. So if your opponent has a lot of other Outwit characters on his team in addition to Apocalypse, watch out