Stemming from a Science fiction story, if nobody minds.
So in ancient locations on Earth we find a gate/portal system, dating back to the Cretaceous. One of those always on, you can look through and see another sky type set ups. Just step across.
A small by-line was after poking around and finding the other side liviable, they try to find out *where* in the Universe they are. And they dragged across a "small radio telescope" in sections, re-assembled it and took two and a half months to determine that they where still in the Milky Way, about 25% further spinward.
Now my ignorance of radio astronomy is rather vast.
Assuming you had all the resources such an event would get you and you were tapped you solve where you were, and given unlimited supercomputer time and exculsive use of the telescope.
I assume not knowing if you were even in the same galaxy would be very influential on the search pattern.
Would that be enough time without being very lucky?
What would be the first markers to look for? Distant quasars or local clouds? I assume they would be distant ones.
About how far from home would you think even a radio telescope wouldn't help you, without at least Earth decades of observation? I would "guess" once out of the Local Group you may as well not bother. (but you wouldn't know that until you looked. )
Or would that still depend on the view? Would you still be more lost if you were a full 50% around the Milky Way, with the full galaxy core in the way over being in another galaxy outward from our side, on the same side, with the same outward view? (I know galaxies don't align up normally)
I'll reword that last one if I have too.