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You can buy the armor from the armorer outside Red Branch Knights' Castle where you land BUT there is also one lying on the ground in the sewers. The ship to Pirate's Island is a sunken ship. So as DadSal says, you have to dive down. : )
The Pirate Queen won't talk to you until you bring her her sharpened sword. Finnu wants the same three gems all over again, a sapphire, a diamond and an emerald. He's greedy. But I should have explained that in a dialogue so you get the bug finder's reward. Just let me know what you'd like AND bring him a diamond, a sapphire and an emerald.
Done. Can buy from superior shops. : )