Similar issues have been asked many many times before, see .fbx export why there are no materials or textures? or Issue with Importing 3D Models into Blender
The STL file format doesn't support textures, its a geometry only file format suited for 3D printing and computer based fabrication processes.
It doesn't include any color or texture representation structures, as far as I know it doesn't even support UV Coordinates.
UV coordinates are what is needed for 'wrapping' a 2D texture around a 3D mode, they can be either automatic, or obtained by UV Unwrapping
There is no such thing as '3D textures', at least not in the way you mention. Those are generally either procedural textures or voxel based textures for volumetric data representation (like smoke or sparse mediums)
If your 3D printer does indeed support color information you will have to look into its documentation what file formats it supports and what it uses for color information. And you will most likely need a separate image file for the texture.
Edit: Ok I think there some confusion and wrong terminology use here hence the misunderstanding from my part.
What you downloaded is not a 3D texture (textures are images with color information), it's a 3D model of a stone pattern albeit one achieve with an image texture through displacement, but that is irrelevant.
If you want to wrap it around in a cylinder shape, forget your STL cylinder, it is useless, its probably easier to just do it from scratch.
Use the .blend file from the download archive, open it, apply Displacement Modifier, then add a simple Deform modifier set to Bend.
Add a an empty object, rotate it -90 degrees in X axis and use it as a origin object in the Deform modifier.
Increase the angle of the deform modifier until it wraps up into a cylinder.
Now it's only a matter of closing the caps and removing doubles so that it forms a manifold closed shape.