Abby, I suspect the word "aboard" may be a corruption of another word that makes more sense, giving the line the meaning that "we're nearing the port"; at the moment, though, I can't think of what that word or words might be. "Hard by", maybe?
And Margo, I believe you're correct. As I read somewhere (so it must be true), the word "starboard" was originally "steerboard" or "steering board", as some old boats were steered with a long oar, or board, that was fastened to the (right) side of the boat, instead of a movable rudder amidships. It would make sense, then, that you would tie up with the other side to the port, so as not to damage the steerboard. Then again, my sailing savvy is rather limited, and somebody like The Ancient Mariner may be laughing in his grog at all this. Eh, Dave?

Aloha,
Mark