The battle has two rounds, the first one against Doctor Cid alone and later against Doctor Cid along Famfrit.
In his first phase, Doctor Cid has 1,000,000 HP and limited attacks, mostly Protect, Shell, normal attacks and Gattling Gun for moderate magic damage to the party and Haste to increase his number of attacks per turn. His buffs must be removed as they give him incredible augments that will make him impervious to damage. Haste cannot be dispelled. Haste and Gattling Gun are used once per battle after reaching 50% HP.
In his second phase he is more difficult, with 2,500,000 HP, Famfrit's company and increased arsenal of attacks. He will still employ Protect and Shell and can attack a minimum of 8 times per turn (12 with Haste). He also employs Anti-Magick (essentially Reflect, it can be dispelled). Every 5 turns he will use Readies tokamak and then S-27 Tokamak the next turn for good Magic Damage to the party as his sole move. Every 6 turns he has a chance to use Gattling Gun. At 90%, 60% and 30% thresholds he will use S-85 Cyclotrone for greater Magic Damage to the party.
If Famfrit is felled he will stop casting buffs and will gain Haste and increased damage on his abilities, making him much more aggressive.
In both rounds Doctor Cid has 100% resistance to all elements except Dark and Light (50%), immunity to all status ailments except Disease (70% resistance) and only susceptible to DEF/SPR Breaks.
Famfrit's attack pattern is much more simplistic, and also has 2,500,000 HP but higher stats. Famfrit can attack up to 3 times per turn, mostly consisting of Waterja for big Water Magic Damage to the party, and Briny Cannonade for major Water Magic Damage to a single target, which always targets the unit with the lowest HP.
While Doctor Cid is alive it will use Readies cannonade and end turn, and shoot Briny Cannonade the next turn as his sole move. At 70%, 50% and 40% thresholds it will use Waterja as its only attack. Once Doctor Cid is dead, Famfrit can use normal attacks and will employ Briny Cannonade every turn without charge time. Also every turn it will use Waterja and end his turn. If Famfrit's MP is drained then it will no longer use any Water attacks.
Famfrit resists Water (200%) and Light (100%) and has a weakness to Fire (-50%), immunity to all status ailments and stat breaks.
The battle is not difficult but will require some preparation. A Dispeller to handle Doctor Cid's buff is a must and Elemental Weapons/Attacks are unwise in this fight unless the player can severely Imperil Doctor Cid's resistances, which will often require strong base 5★ units like Orlandeau, Aileen, Fryevia or Marie. Marie doubles as a solid buffer for this fight, but she's ill advised as her buffs, except Love You All are Green Magic, preventing the player from achieving one of the missions.
Xon is also an interesting unit to use as he can steal Doctor Cid's massive buffs for the party making them very much invulnerable to damage, although this will only last for a rather short time of two turns. Apart from this, he will have limited use in the battle.
Minfilia is a good buffer as she can buff stats for several turns and has a 70% Water Resistance buff. None of these abilities are Green Magic, making her more useful than Marie.
Healers like Y'shtola, Refia or Fina are good choices. Luka also doubles as a buffer with Water Resistance, but it is more advised to keep the roles separated for efficiency.
Ling is a useful debuffer as she can inflict Disease with Dance of Death and reduce Doctor Cid's already pitiful defenses even more allowing him to be taken down easier. Warrior of Light is also useful as he can Tank attacks, inflict Debuffs and has a chance to regenerate his HP and MP as he's attacked.
There is ample room for attackers as both Doctor Cid's and Famfrit's defenses are low. Elemental attacks should only be considered if one can build a high elemental chain count (i.e. 2 Fryevia, Orlandeaus, etc.). Since one of the missions is to use Waterja, the player will likely need a Mage unit. Zyrus and Dark Fina are ideal choices, the former can use Blood Pulsar for non-elemental damage that grows stronger and Flare against Famfrit. Dark Fina with enhanced Ultima can deal decent damage.
Unlike in Final Fantasy XII the player is free to decide what Boss to take down first. Famfrit is the easiest of the two, so he should be left last, as Doctor Cid will become significantly more difficult to handle if left alone. While Doctor Cid will take more time to defeat as a first approach it will make the battle the easiest in the long-term. If the player is confident they can defeat Doctor Cid, regardless they can choose whoever they want.