Golbez' behavior changes according to how far the player is in the battle. He has 3,800,000 HP, modest ATK, MAG, decent DEF and high SPR.
During the start of the battle and until he is below 50% of his HP he will use a very simple moveset, consisting of Firaga or Thundaga once, to damage the whole party, or Bio up to 3 times to inflict Dark Magic Damage and Poison (100%) on its targets. Regardless it is followed by normal attacks, which will cease to be from phase 2 and onwards.
Once he is below 50%, he will summon Shadow Dragon which also reduces the party's HP to 10 and end his turn, however a cutscene plays where the party's HP will be restored to half their max HP (this cutscene mirroring that of Rydia helping Cecil on the original game). With the Shadow Dragon up, Golbez will resort only to his above Spells in addition to Drain. He has a chance to select between Drain, Drain or Bio, Firaga or Thundaga, Bio or Drain.
The third phase plays when the Shadow Dragon is felled Golbez will use a random attack at the beginning of his turn. Of his entire skillset it includes 3 chances to use Binding Cold (including summoning Shadow Dragon which won't summon the Dragon but will still reduce the party's HP to 10). This only happens once per battle, but it can be followed by his new arsenal of attacks, being Osmose, Firaga or Thundaga, Osmose or Bio, Drain or Firaga or Osmose or Binding Cold (with added Paralysis). Which means there is a chance for a party wipe. His Osmose's expect for the third one of his list, will target the unit with the most MP.
After passing that turn, Golbez will move to that moveset although Binding Cold's chance is now only 20% per turn. There is however a small chance he may use it before his last action, in which case it can lead to a party wipe.
Golbez has no resistances or weakness to any element, is immune to all status ailments and immune to all stat breaks.
The Shadow Dragon has 3,200,000 HP, it has decent DEF at 200 but poor SPR at 30. The Shadow Dragon has only one attack, his signature Black Fang which deals fixed 999,999 damage to one target thus ensuring it is KO'd. It will do nothing during the turn it is summoned however.
Black Fang will be used once every turn, but if Golbez is killed first while the Shadow Dragon is not, then Black Fang will strike all units at once, and will repeat this every turn.
The Shadow Dragon has no resistances or weakness to any element, is immune to all status ailments and vulnerable to all stat breaks.
Stat-wise, the battle encourages the player to use both physical and magic damage for both Bosses, but it is not needed as decent characters from either ATK or MAG skill can defeat the both of them, although the player can diversify for an easier time nonetheless.
What the player has to contend with almost total certainty is death as both Black Fang and Binding Cold are flags for destruction. The most efficient way to deal with this is with Reraise, the best one found in Rikku's Limit Burst which provides Reraise to the whole party. Given Black Fang doesn't specify a target this will ensure the party is defended, even from some Binding Cold wipes. A Tank with Provoke-esque abilities can serve as a sacrificial pawn for Black Fang, although Reraise would still be needed to keep repeating this tactic.
Golbez's attacks aren't that strong but with elemental protection and SPR buffs, the damage should be quite manageable. Most likely the Shadow Dragon will be the biggest annoyance in the fight. To kill it quickly the player can capitalize on its low SPR and finish a chain on the Shadow Dragon with a strong Magic Attack. If combined with passives that boost Magic Damage on Dragons such as those found in Zyrus and his Draco Spike, then inflict massive damage on the Dragon should be easy. Stat Breaks can further capitalize this.
Given both Bosses have solid defenses in one particular stat, using chains will be effective for both of them, however the player must prioritize destroying the Shadow Dragon first to prevent Black Fang to strike all the party. Given neither has elemental resistances, the usual chainers like Orlandeau, Veritas of the Dark, Fryevia, or any that can also inflict elemental debuffs, can easily deal a lot of damage.
Fortunately for the player the 5 or less man-party mission yields a 5% Trust Moogle which while useful is the most trivial of the rewards and easily acquired through multiple means, so if the player has difficulty in beating the battle with reduced members, they can use a 6-man party and suffer minimal setbacks.