Hello to all my fellow new (or at least new-ish) pirates out there, and a hearty Yarr to ya me maties! I've been practicing a fairly simple set of skills to dock relatively quickly and easily even as a solo sailor, and I thought I'd share my results in case others would like to give the method a try.
The first real "trick" here is that you can grab your anchor while it's still falling to the sea floor. When you do this, it stops more or less instantly, so if you remember how far the capstan turns before actually stopping the ship (I don't know for a galleon, but for a sloops it seems to always be one full 360 degree rotation) and stop it right before it grabs the bottom you can then release when your ship is exactly where you want it to be and stop almost instantly. As a solo pirate, it may help to raise your sails partway up during your final approach, giving you a bit more time to aim your ship right where you want it. The idea is to have the very front tip sort of dangling over the dock, so you can run from the prow to the dock and back without having to swim at all. Now you may be thinking, "but how do I get back out of there? as soon as I drop my sails I'll ram right into the dock." The "trick" now is to leave all your sails all the way up, turn your wheel all the way to one side, and raise your anchor. This way your ship should almost turn in place. Pick which direction has an easier path away from the dock, turn that way, and when your ship is parallel to the dock drop your sails and get underway.
I hope at least one or two new players find some helpful information in here. Cheers and thanks for reading!