Is there another way to discover it, other than completing 4/4 quests for the Don?
There are actually two ways in to the passage, both via secret doors. But then you've got to get out of the passage to the outdoors and that requires a key.
If all you want to do is get out of town and you have an acrobatics ring (I really should get trained in that, but I've always got an acro ring on so nobody ever offers it), you can jump off the walls. The west wall has a ton of guards but it can be done. The east wall is much more convenient, IMHO. Here's a quick video showing how to get out. I don't know if you have to pay again to get back in.