Two ways to do this, either the typical route or the fast route.
Typical route: Completing the Obsidian Pinnacle on the Default difficulty will unlock the Adventure difficulty, and so on with Adventure unlocking Apocalypse. From here, you will have access to the Apocalypse difficulty modifier for any level at that point, though you might find that they are considerably harder than the previous difficulty. No new mechanics are often added between the difficulties, but instead more mob spawns or more enchanted mobs in general will appear.
Fast route: If you have a friend who has completed the difficulty unlocks and has frequently played in Apocalypse difficulty, they can invite and carry you through the level. You do not have to participate directly in any combat pieces. The only thing you'd have to watch out for is if anyone else, or you, dies during it, as when night comes, the added enemies might get you from your hiding spot.
The fight against the Arch-Illager remains the same. You will have two phases, where the first will be him teleporting about, summoning a few mobs here and there, and hitting you with ranged attacks. Once you've depleted the health bar entirely, the Orb will embody him with the power of the Heart of Ender. In this form, the Arch-Illager's body is still the target to attack, but he will be more chaotic in his attacks. From spewing out dark energy similarly to the Redstone Monstrosity from the Fiery Forge, to diving into the ground and spawning four lasers, to even performing a rotating laser spread that you can't roll out of without Ghost Form; all of these attacks will remain the same, just become stronger in potency.