A very good first draft. I like how quietly Jack snuck aboard the ship and stole the treasure chest.
It looks like there are plenty of places where you can cut out 7 words. Just go over every sentence and see where you can trim a word or two and soon it'll be under 300 words. I'll point out a few places that I think would really help make the flow better. Also, here are some corrections. Please feel free to take them or leave them. I'm just trying to help. :-)
whilst Captain Brickbeard was enduring another restless, cold an uncomfortable sleep imprisoned in the ship’s hull.
I believe the word "an" is supposed to be "and"
The Governor was dreaming sweet dreams of capturing more Pirates.
Here's a place you can cut out a few words and get rid of the repetition. I suggest removing "sweet dreams" and just have the governor dreaming of capturing more pirates. Or, if you like sweet, you can say he was sweetly dreaming of capturing more pirates. It takes out one word and the repetition is gone.
But unknown to the Governor or any of the crew of the Imperial Flagship,
I'd suggest getting rid of "But" as its not a good idea to start a paragraph with it, but also because it doesn't really add to what you're saying.
...there was a pirate who was much worse than Brickbeard, who was the most feared pirate at the time. This pirate went by the name of Jack Sparrow, Captain Jack Sparrow. Who was currently unknown to the Governor even though he was in the same room as him and stealing his treasure chest.
You've got a lot of words here that don't need to be here and there's some repetition. You also have some fragments, though I'm not opposed to a well-placed fragment, it's usually frowned upon. Also be careful of using too many pronouns (he, she, etc) in one sentence as readers may get confused as to who you are talking about. Here's what I'd suggest:
...there was a pirate much worse than Brickbeard, who was the most feared pirate at the time. This pirate went by the name of Jack Sparrow. Captain Jack Sparrow was currently unknown to the Governor, even though Jack was in the same room as him and stealing his treasure chest.
He made his way back out to the deck of the Imperial Flagship, which was currently connected to the hull of the Black Pearl, a ship looming over the comparatively small Imperial Flagship.
This is a bit of a run-on sentence with more words than it needs. Here's my suggestion
He made his way back out to the deck, which was currently connected to the hull of the Black Pearl. Jack's ship loomed over the comparatively small Imperial Flagship.
After another few seconds of climbing Jack and the treasure was being helped on to the deck of the Black Pearl.
There needs to be a comma after climbing.
He now gave a long piercing whistle, a signal to set sail in an unusual direction, downwards.
You don't need the word now, especially since it changes the tense of your story. I'd also suggest some alternate punctuation. It's not necessarily right, but it may have a bigger impact.
He gave a long piercing whistle; a signal to set sail in an unusual direction--downwards.
Jack completely knew the effect of that whistle within a few seconds he saw the desired effect he could see the confused face of the Governor at this cabin window.
You need some punctuation in this sentence, or you need to split it in two. And you have too many words saying the same thing. Also, the word "this" should be "his." Here's what I suggest:
Jack completely knew the effect of that whistle and within a few seconds, he saw the confused face of the Governor at his cabin window.
The Governor now had a story for the grandchildren, a pirate with pink skin stole his treasure and sailed off on a massive black ship down in a great wave to the depths of ocean.
Since this is from the Governor's perspective, I suggest putting this on a new line. It will also punctuate the end of your story. And instead of the comma, I suggest using the double dash.
Hope my comments help! Good luck in the contest!