Texture Atlas applied to existing mesh?

I'm not sure if this is the best forum to ask this in, however I'm not really sure how to actually do this.
Prior, I have messed with texture atlases on generated meshes. However, now I need to offer a range of colors on a existing cylinder mesh. I tried to use random color generation, but that caused the draw count to skyrocket.

What would be the best way to use an existing mesh with a texture atlas? My goal is to create something like this:

However, without using 'color' - which is how the various colors there were generated.