How long after buying a boat can you legally change your name to Cap'n?

Change it to Admiral straightaway. Much more impressive.

Who ever is in command of a boat is the captain of that boat or ship. For example in the navy the commanding officer of a submarine is called the captain, not matter what the rank is of the officer. The Coxswain on a fifty foot motor boat in the navy is also the "captain" of that boat.

Now there is also a "rank" of captain, which has to be earned. Unless you join the merchant marines, coast guard or navy and earn the rank of Captain, you will not  be qualified to be a "Captain".

For what it is worth this carries over into the aviation also. The person in charge of the flight (which may or not be the pilot) is called the "Captain".

So there are two uses of the word captain, one is a position of authority and the other is an earned rank.

Please note that once the "captain" leaves his boat, he is no longer a captain when used as a title of position of authority. For example when the Captain of a USN ship gets on to his gig the coxswain at the helm is the captain of that boat. Once the coxswain steps off that boat on to the ship or onto shore he might be a seaman recruit. But the officer who has the rank of Captain remains a Captain no matter where he is.

So when you are on board your boat, you are the captain, provided you are accepting all the responsibilities that go with being a captain. But once you get off that boat you are just you.

You don't legally need a boat at all. By procedures of location law courts, if a person wants the name Captain there are no laws against it. The USA has no legal titles. You can change your name to Emperor if it pleases you.