It's an informal phrase that isn't necessarily grammatically correct. "I will take first watch" would perhaps be the correct way to say it, in future tense. People often say "I got" to insist on doing an action themselves so if they insist on paying for someone in a restaurant, they would say "I got it." even if they haven't paid yet.

It's an idiom that means I take responsibility/I can deal with/I can manage this.

Grammatically wrong, it would be fixed by saying "I have got first watch" in which "got" serves as past participle to the verb "to get". It is, however, a common colloquial way of speaking. Just imagine the missing word "have" to be there, and it will make sense.

It sounds like a colloquial way of saying ‘I’ve got’.