You can see little black lines, or do I really need another program for that?
You’re talking about the CRT look, the old television set look? After Effects would work great for that. You can’t do that in Blender, because it’s not an object nor an object render- it’s a video effect done in post-processing. This tutorial on making an old television effect in After Effects might help, but if not, I’d go to YouTube and find another one there.
If you don’t have After Effects, there are free “solutions”. I would not recommend using them.
Is Adobe after effects better than Adobe premiere pro for adding effects to videos?
In my experience, yes. You can technically use either, but After Effects has a lot of effects that Premiere doesn’t. The two work well together- Premiere is a much better video editor, After Effects is better for Effects, using both usually gets most video work done. In this case, you can probably get away with just one or the other, but After Effects will probably work better for that particular effect
The effect can be roughly approximated in the compositor.
This would work best for a still render, if you wanted to render an animation you might want to put more effort into the effect.