<p>I was thinking about <a href="https://lifehacks.stackexchange.com/questions/24627/how-to-paint-a-circle-pattern-on-a-big-piece-of-paper">this question</a> and thought of maybe printing a pattern-drawing roller painter.</p>
<p>The question is: is it possible to print with an ink absorbing material that could make a paint roller possible?</p>
<p>P.S: I don't own a 3D printer, nor have I any deep knowledge in this matter. I simply want to know if this is feasible, so I can start looking for someone to 3D print this for me. If it's not, knowing beforehand could spare me a lot of time.</p>

Not really.
It's possible to 3D print in materials like TPE that are rubbery. In theory, one could print a sheet of the pattern and then wrap that sheet around a roller. That would be expensive, though, and I doubt TPE would absorb enough paint/ink to lay down an good and even coat.
As pointed out in the comments,there is one of foaming TPU filament (
Varioshore TPU
that might be able to achieve the kind of soft, spongy feel that a paint roller would need but it's expensive and, I suspect, soft but not particularly absorbent in the way you would need.
I have no direct experience with it, though, so I can't say for sure. For the price and amount of time that would be required to get a print roller produced the way you need, I think you'd be better off buying a
custom pattern roller/brayer
or making one yourself.
I imagine you could achieve this by getting a sheet of stamp pad foam and plotting it with a CNC cutter (or cutting it by hand) and then taping/gluing it around a tube.