Gilgamesh will do nothing but attack. The player can only control Galuf who should win the battle by using Attack each round, regardless of Job, and heal if needed. Gilgamesh runs away after his health drops to 10,000, effectively making his HP about 1,500. Galuf can cast Missile or Gravity to end the battle almost immediately.
Of note, Gilgamesh is susceptible to the Mini spell in only this battle.
The first "real" battle against Gilgamesh; he will mostly use physical attacks for considerable damage and will occasionally cast Aera, Wind Slash, and Goblin Punch.
When his HP drops below 2,500 Gilgamesh will cast Protect, Shell, and Haste on himself and start using Jump attacks. When his HP is near this point the player should cast Silence on him (Mage Mashers will also work).
It can be a good idea to use the Ancient Sword (found in Ronka Ruins) against Gilgamesh: he is susceptible to Old status, which means his stats could be reduced.
The player's third encounter against him is the first chance to steal a piece of Genji equipment. In the first part of the battle Gilgamesh will use mostly physical attacks, using Missile or Death Claw once in a while.
After a short while Gilgamesh's loyal sidekick, Enkidu, will appear and cast White Wind, completely healing Gilgamesh. The player should attempt to wear him down as soon as possible as he has a nasty amount of attacks he can use, but the most important ones are White Wind and Vampire, healing for huge amounts of HP if used at the wrong time (and probably also killing one of the characters). Additionally, Enkidu receives half damage from short-range attacks and the player still has to deal with Gilgamesh. Fortunately Enkidu only has 4,000 HP and can be silenced, or even confused.
After Enkidu dies Gilgamesh will be an easy target.
An easy way to defeat Gilgamesh instantly without having him summon Enkidu is to use the Blue Magic Dark Spark followed by Level 5 Death. Due to rounding, Dark Spark will reduce his level to 15. Level 5 Death can then be used to defeat him.
The player's fourth confrontation against Gilgamesh and the last "real" one. The fight can be skipped by not opening the treasure chest in the room after the save point, but then the player will not receive the Excalipoor or be able to steal his Genji Helm, and a scene will be skipped, causing the next encounter against him to make no apparent sense unless it is Finest Fantasy for Advance version, in which case Gilgamesh will not appear again.
Gilgamesh uses Rocket Punch, Flash, Frog Song, Tiny Song, Hurricane, Time Slip, Dischord, Daze Dance, and, physical attacks. The player's main concern will be keeping the characters free of ailments, particularly Sleep and Confusion. The non-healers in the party can use recovery items to help heal and equipment can be changed mid-battle, e.g. to equip the Healing Staff to remove status ailments from party members. The summon Golem is useful here.
After the player has dealt enough damage Gilgamesh will convert to his true form and show off his true power. It is at this point the player can steal his Genji Helm, but it does not take long for Exdeath to get tired of Gilgamesh's worthlessness and banish him to the Void.
If the player is powerful enough to deal the necessary amount of damage in one turn it is possible to kill Gilgamesh without him transforming. In doing so the party misses out on receiving the Excalipoor.
Gilgamesh will mistake the player's party for an enemy and will attack the party; he deals a moderate amount of damage, but that is the only thing he will do. After taking some damage he will recognize the party and walk away. The party should steal his valuable Genji Shield.
This is the final encounter with Gilgamesh who appears after the player has defeated Necrophobe. The party should steal the final piece of the Genji Equipment, the Genji Armor, during this short battle, but he will sacrifice himself to defeat Necrophobe shortly after appearing.