Is there proper horse back riding etiquette rules for the arena ring?

There certainly are! Here are some of the most basic rules for an arena.
When mounting your horse, either mount outside the arena or in the middle, if someone is riding in the middle, simply mount in an out of the way place.
Do not ever lunge a horse in an arena when there are others riding.
Do not ever tie a horse inside the arena, most public arenas have other places that are suitable to tie horses at.
Do not use up the whole arena, I know this sounds like a no-brainer, but you would be surprised at how many people come into an arena that someone was using first and hog the whole thing.
If you and another rider are riding in opposite directions and need to pass each other, pass so that you are left side to left side, so your left shoulder and their left shoulder will be next to each other while passing, this rule (left side to left side, or left shoulder to left shoulder) will let you know who needs to stay close to the wall and who needs to move their horse over.
If you are riding in the same direction and you need to pass their horse, do so on the inside (closer to the middle of the arena) and say, somewhat loudly, something along the lines of "Passing on the inside".
Make sure that you know the rules of the arena that you are riding in.
If you follow these rules closely you should be good to go :-)