My guess is you're writing a fiction story, in which case you might as well just make something up because 99% of your readers won't know the difference and the 1% that does won't even care.
This is a scientific concept so esoteric that you can literally write it however the plot needs it to be and you'd get away with it just fine.
On the contrary, science fiction has a lot of readers who are scientifically savvy and who appreciate it when writers take the time to do their research, and who notice when they get it wrong. There's a whole subgenre of hard science fiction that's all about getting it right, or as close as possible, and readers of that genre certainly
do care. If you're writing for that market, then you're definitely better off doing the research.
If you set 0 degrees heading to be the direction the galaxy is headed towards "The Great Attractor" which is itself out towards the Hyrda constellation, and this area itself lies within the "Zone of Avoidance" what degree heading would the earth be in such a coordinate system?
I'm thinking it would be somewhere on the backend of the galaxy around 180 degrees but does someone with more knowledge have a better guess?
I can't fault your reasoning. The galactic center's in Sagittarius, so Hydra and Centaurus (the area of the Great Attractor) would be just a bit north of the galactic plane and a little off to the side.
It's tricky to get specific, though. Since we're moving along with the rest of the galaxy, it's not easy to compute its direction of motion precisely, and there's some uncertainty there.
The best estimate I can find is that it's toward the Norma Cluster (Abell 3627) at RA 16h 15m 26.7s and dec -60° 53' 17". We can treat the difference in angle between our POV and the galactic center as trivial on this scale. The galactic center is at RA 17h 45m 40.04s and dec -29° 00' 28.1". That gives an
angular separation of roughly 35.2534 degrees, or 35° 15' 12.24". But if we look at it from the galactic center's POV, we'd be in the opposite direction, so we need to subtract that from 180°, giving 144° 44' 47.76".
That just tells you the raw angle between the Galaxy's direction of motion and the vector connecting earth to the galactic center, and I'm not sure it would be enough to specify its position with only one angle. You'd need to split that into a right ascension and declination relative to your coordinate axis. But as it happens, the Norma Cluster is just about in the plane of the Milky Way -- actually
about 7 degrees off the plane. I'm not really sure how to use that to correct that figure above; I think the angle within the galactic plane would be a little bit smaller, but I'm not sure of the formula. But 7 degrees is small enough that it wouldn't make too much difference, and there's a pretty large error in the actual galactic direction of motion. If you gave it as something in the ballpark of 140 degrees (or 9 h 20 m) by 7 degrees, that might work.