The patch is here and Small Time Buccaneer is nerfed. So what's the state of Pirate Warrior? Obviously the early game is far worse than before, and that's the weakness of the deck now, it's much harder to get good draws early, but that leaves room for some stronger cards later like Naga Corsair and Southsea Captain, so even though you no longer have a god tier 1 drop the deck is still very viable.
But who am I to talk all big and mighty? I might not be an amazing player, but ever since beta I've mained Warrior, with two small tournament wins playing mostly Control Warrior. So I've played every Warrior archetype there has been, during every patch. So trust me when I say Pirate Warrior is NOT dead, and will probably continue to sit among the most played classes.
Mulligan is very similar to before, you just have a bigger focus on getting 1 or 2 drops like Southsea Deckhand, N'Zoth's First Mate, Sir Finley Mrrgglton, Bloodsail Raider and Fiery War Axe. Also keeping a Dread Corsair with a Fiery War Axe is a really strong start, especially with the coin.
Edit: Hobart Grapplehammer is probably a good card to run since the buffs he gives are very strong and synergise very well with the deck, and would help a bit with the early game. I just don't have the card, because I don't want to craft a non-golden version (I have almost every Warrior card in gold) and therefore don't play it myself. It is definitely not needed for the deck to work, but if I had it I would replace it with a Naga Corsair or Mortal Strike
Post KOTFT Edit: As much as I hate the deck at this point, it's probably still viable. Replaced Sir Finley Mrrgglton with a second Southsea Captain, but you could put in Hobart Grapplehammer, Acherus Veteran, Bloodsail Corsair or almost anything else. It's almost unchanged, and it's as braindead as always. (Actually even more braindead, since you don't get to choose a new hero power). Looking forward to next year when the cards rotating out hopefully kill pirate warrior once and for all.