Well, this deck should be more resilient than the logical pirate version with the quest.

More control to gain armor and enough control to remove threats.

Mulligan: pretty simple as one card is your quest then:

Vs aggro decks: keep/search your pirates 1-2 mana like Bloodsail Deckhand/Harbor Scamp and removals like Shiver Their Timbers! (rancor can also be considered), reject your weapons as you have only 2 and you will draw them with the quest

Vs control decks: think about completing the quest asap so pirates are key

Middle game: continue to put some pressure and remove threats, you should have your quest ready. Use Whatley if you have it asap to ensure maximizing your draw

End game: enough armor, use Alexstrazza to finish the game or N'Zoth. Before using N'Zoth, try to play all your main minions like Alex, Kresh, Golem...