You should be able to get finer than 1 mm with the Omni 4000. I just started using mine, and had little trouble getting that fine. This airbrush has a dual taper needle, allowing for both coarse and fairly fine work with the same brush.
First thing: take off the air cap guard, reverse it, and put it back on. This allows you to get closer and changes the air flow pattern at the tip.
What follows works for me:
Make sure your paint is adequately reduced (thinned). For fine lines, thinner is better.
Reduce your air pressure to 8 to 10 psi.
Practice, practice, practice!
A final tip: I've discovered that you can adjust the trigger resistance to suit your fingers by adjusting the spring adjuster—helps a lot with control.