Materials for research on spherical projection
You need to use a cylindrical projection. So the key issue is how to prepare data for a cylindrical projection. Basically what you do is: 


You project your data on a sphere, it may be that this process was already done for you.
And then you project it back to a cylinder


Both steps can be easily taken care of by your cartography software. Will the data distort... yes but since you project information onto a sphere from whatever source onto a sphere and to cylindrical coordinates it will just interpolate the distortion away. If anything your just going to have slight over resolution at poles.

Quite a lot of data is in fact in this format to begin with. But nearly all 3D applications can use projected textures and texture baking to do these steps for you. Also most 2D applications like Photoshop have tools for turning rectangular coordinates to polar coordinates which do the same thing

The real question is what your original image data is like?