Hey, shrinkwrap works best if the object youre projecting on already has a similar shape, so it doesnt have to project from such a distance. In think in case of projecting on a cylinder, you can use simple deform modifier and set it on bend. If you align it so its straight up, you can bend it around z axis to get a shape similar to the cylinder.
Then you can fiddle with it until its alligned if you want. Alternatively, you can delete one side of it, so its only a flat (actually a bent plane if you already put on the simple deform) plane, and then shrinwrap it around the cylinder to get it perfect. After that you use the solidify modifier to get the thickness back. You do that because shrinkwrap would project both side.
You can also not do any of that and create a bezier circle, and then use curve modifier and set it on the circle. If the circle and cylinder are concentric, you can just scale the circle until it matches.
