I agree. Since the First Officer is technically second in command, and probably received commendations for getting the ship and remaining crew out of a fatal situation, they were probably promoted to Captain of the ship. Since Kirk showed heroism above and beyond in saving the ship and crew, it probably warranted such a promotion to First Officer of Farragut.
Where is there any indication that Kirk did anything heroic in the Farragut incident? He blamed himself for hesitating when the Cloud Creature killed Garrovick, and Kirk was always brave and acting bravely. But I never remember seeing or reading anything to indicate that he was somehow instrumental in 'saving' the ship (which was structurally unscathed, probably) or saving what remained of the crew.