Larry Niven kind of works out this naming in several of his novels. I don't remember all the specifics, and he also used a similar scheme to describe travel in ring world, but it's close enough. First, don't bother with calling it north, that is just confusing. In the reference frame of yourself or the map you're drawing in a spinning galaxy, you've got spinward (in relation to the galactic spin) and anti-spinward, in (toward galactic center) and out, and then normal (orthogonal) to those dimensions, which you could call up and down depending on your preference. I'd probably call spinward, inward, and up positive.

If you need a reference (north) for a galactic map, it's really unlikely you'll want to use anything like grid coordinates. You can use the same ideas from the local map. You'd probably want an origin at the gravity center of the galaxy and pick another object as a reference point from which to zero angular measurements around the disc. That other object could be another galaxy (if you want to measure galactic spin itself) or something distinct and obvious in our own galaxy (if you want to navigate within the galaxy). Most civilizations would probably just use a line between their home system and galactic center as their prime meridian. Up and down should be orthogonal to spin again. If you're home planet had a magnetic pole roughly pointing out of the galactic disc (like ours), you'd probably choose your "north" pole's side up.