If you cast say Mythos of Snapdax using the alternate Jodah, Archmage Eternal cost, will it activate the boosted mode of the Mythos for having cast it with the required colors?
Yes. If you cast Mythos of Snapdax by paying WUBRG rather than its mana cost, then for the purposes of that spell, BR will have been spent to cast that spell, even though its mana cost wasn't paid this way.
In general, for the purposes of things that care whether certain mana "was spent to cast" a given spell, what is relevant is what mana was spent to pay that spell's total cost, not just the mana cost or an alternative or additional cost of that spell (review C.R. 601.2f, 601.2h). (However, if the spell is copied, the copy will not have been cast, even if the original was [C.R. 706.10], so that the condition "If [certain mana] was spent to cast this spell" will be false for the copy.)
In general, for the purposes of things that care whether certain mana "was spent to cast" a given spell, what is relevant is what mana was spent to pay that spell's total cost, not just the mana cost or an alternative or additional cost of that spell (review C.R. 601.2f, 601.2h). (However, if the spell is copied, the copy will not have been cast, even if the original was [C.R. 706.10], so that the condition "If [certain mana] was spent to cast this spell" will be false for the copy.)