You simply can not prepare an exact two dimensional file for a Sphere. This has being the problem for making acurate maps of the earth and visual apealing at the same time.
Here is a recopilation of diferent maps trying "to solve" the problem: http://www.otake.com.mx/Apuntes/Imagen/EnviromentMaps/Maps.html
What you can do is a projection.
What type of projection is the issue.
Here is an article of wikipedia: https://en.wikipedia.org/wiki/Map_projection
And here is an essay I made on the most common projections for 3D environments, but can be used for your case: http://www.otake.com.mx/Apuntes/Imagen/EnviromentMaps/
For simple spheres, like a billiard ball you can use a planar projection or cylindrical one.
For a practical aproach, the design should meet the system used to print the object.
For example it can be printed rolled on a silk screen. Then you need a cylindrical projection.
It can be delimited in the 3D pice itself, so you can construct it in a real 3D model, or it can be printed in an hydrographic system, so it can be a planar projection.
Once defined the type of projection you can prepare a 2D design on Ilustrator.
Preparing a texture for a full sphere is another thing. A texture needs to be repetitive and in a pattern. To greate a pattern you could base the design on a triangular face and use a "geodesic" aproach. In a 3D model, for example in blender is called an Ico Sphere.
You also can base the pattern in other shapes, like the pentagons and hexagons of the old style soccer ball.