AtmaWeapon
Speaking to the monstrous form on the Floating Continent triggers a battle against AtmaWeapon, the most intense boss fight yet. It even has unique music to set the mood. AtmaWeapon has a massive 24,000 HP, and no weaknesses to speak of. Its basic attack is frighteningly powerful, but less dangerous than Flare, which it will also use regularly for about 700 damage. Its other basic attack is Blaze, which is akin to Fire 2. The beginning of the fight will be a mix of hitting AtmaWeapon hard and healing up from its attacks.
Once you've dealt 11,200 or more damage, AtmaWeapon begins a new attack sequence. It starts with Bio, Quake (which will miss AtmaWeapon due to its Float Status), or Meteo, which is then followed up by either a normal attack or the double-damage Full Power special. In the next round it will hit the party with Fire 2, then finish up the sequence with Mind Blast. This causes randomly chosen Statuses to randomly chosen party members, and can be anywhere from ineffective to devastating, depending on what gets chosen for whom.
After Mind Blast, AtmaWeapon gets Safe, Shell, and Haste and starts to flash and glow yellow. You'll have a few turns before it unleashes Flare Star, which deals damage based on your characters levels. It will most likely be less dangerous than many of AtmaWeapon's other attacks. It will continue with this sequence until it is falls below 6,144 HP at the end of it. At that point, AtmaWeapon will start using Quartr, Rasp, W Wind, Blaze, and basic attacks until the fight is over. It will also start using Flare as a counterattack. If you've gotten this far, be prepared to use Cure 2 liberally and keep the pressure up to end the fight.
Escape
When AtmaWeapon falls, Shadow will leave the party. Head up the stairs to find Gestahl and Kefka. If Celes isn't already in the party, she will show up now and join. A host of dramatic events will unfold, after which the party will be in a new section of the Floating Continent with six minutes to escape. The Charm Bangle and Sprint Shoes can be helpful here, although you should have more than enough time to do what you need to.
The only random Encounter you'll face in this area is with a Naughty. They have 3,000 HP and weaknesses to Fire, Thunder, and Pearl. You can't run, so hit them hard and fast to continue on. Physical attacks are generally less effective than Magic unless they ignore Defense, but the Naughty may counter Magic by inflicting Mute on the caster. An alternate method of dealing with Naughties is to cast Imp on them. If you do, they will cast Imp to restore themselves, then flee from battle. They prefer Ice-based attacks like Cold Dust, Ice 2, and Blizzard.
The terrain will crumble beneath you as you proceed. You'll see a blue orb as you go, but you have to be careful to obtain the Elixir inside. Circle around south of the gap in the center rather than heading directly for the staircase. Shortly after this, you'll see a glowing point that turns out to be another boss.
Nerapa
Nerapa has even less HP than the Naughties you have been fighting up until now, but it does have an ace in the hole: auto-Reflect. It is vulnerable to Ice, Thunder, and Pearl, and if you have non-Reflectable attacks of these Elements available, they will make short work of Nerapa. You'll have to act quickly, because Nerapa starts the battle off by using Condemned on the whole party (which Strago can learn if he's present). It will follow up with Fire attacks including Fire 2, Fire 3, and Fire Ball. Fire 3 in particular can be devastating.
If you still have a lot of time left on the clock, and you have Strago along, you can try to learn Roulette here as well. Nerapa will use it after being attacked with Fight six times. With only 2,800 HP, it's entirely possible six Fights will end the battle, but if you use weak (or no) weapons it may be doable. Nerapa will also sometimes counter Fight with a physical attack of its own.
Saving Shadow
Once Nerapa is defeated, you can jump to the Blackjack at any time. However, Shadow is still on the Floating Continent. If you have more than five seconds left, you can wait for him. He will always appear with 0:05 on the clock. If you don't wait for Shadow here, he will truly die and you cannot recruit him in the World of Ruin.
Once you jump to the airship, you'll witness the full effects of Kefka messing with the Statues. The world as you know it is gone.