It might be possible to directly print something like this, however it might not be the easiest method. Let me unpack that a bit.
The issue for 3D printed foams is that, even the low density foams, such as the ones from ColorFabb, still are fairly high density on the scale of foams. (they are about 50% as dense as solid filament when expanded. They are also great materials, side note, just not for this application) They are also what is called "closed cell" which means, while they have hollow spaces inside them, those spaces are closed off from each other. This makes them not very useful for things like holding paint, because the effective surface area is not substantially different from a solid part.
There are alternatives, materials out there which print solid but contain a fraction of soluble material. So you print, and then post process to dissolve away those elements, resulting in a high-porosity filament. Something like Poro-lay might work. (you can find it for purchase HERE)
There might be an easier path here though, and that would be to use 3D printing as an intermediate step. 3D print the design of the roller you want, and use it as a mouldmaster. Pour a silicone or urethane mould material over it, making yourself a mould into which you can then cast any of a wide variety of different rubber or foam materials to make your roller. 3D printing as an intermediate step, rather than end-use-part, is very very powerful in that way.
With all of this, chemical compatibility between your paint and end material should be kept in mind. Some paints contain solvents which will attack various foams, plastics, etc.
Finally, while this is digressing from the realm of 3D printing, there is one more solution I might consider exploring. A C02 laser will cut, or engrave, the felt on most paint rollers. If you can find someone with a rotary axis who is willing, they can simply cut/engrave the felt/foam of a factory paint roller into whatever pattern you like. (this is how the CUSTOM RUBBER ROLLERS linked in @Rykara's post were made) It wouldn't actually be terribly difficult, however you'd probably want to experiment on a couple different rollers to ensure the effect is what you want.
I hope that helps. :)