Your first fight with Ultimate Weapon is when he attacks you at Mideel. After
a while he runs away, but whatever damage you do to him will stay. Later on
you will be able to fight him again. He will be waiting over a small lake
of water, so just ram him with the airship, thus initiating the battle. If
you manage to kill him, after all he only has 80,000 HP, so one KOR can kill
him, he will go straight to the area where the Ancient Forest is, right next
to Cosmo Canyon. There you just do a bit more damage and hes gone. If you
didn't deplete all 80,000 HP, then he will run from battle and fly around
the entire planet. You must chase him immediately, because if you don't, theres
a pretty good chance that you'll lose him for a good chunk of the game. If
you chase after him, then you must ram him three times with the airship to
get him to go to the next place where you can fight him, above Neibelheim.
Then the process keeps going and going until you do deplete his HP.
No preperations are really needed. Just equip your most powerful equipment and best materia. If you have Knights of Round then you can kill him with just one casting and then one attack over Cosmo Canyon. Thats pretty much it.
During the battle, Ultimate may just pick up and leave after you've done substantial damage to him. His attacks are all semi-powerful. When you kill him over Cosmo Canyon he will use an Enemy Skill called Shadow Breath on you. This may, or probably will, kill the character on whom it is cast. Its very powerful, so make sure each of your character has an Enemy Skill materia, that is, if you want it.
Your reward for beating Ultimate
Weapon is Cloud's Ultimate Weapon, the Ultima Weapon, as well as Shadow Breath
and access to the Ancient Forest without having to use a Green, Black or Gold
chocobo.