Painting acoustic drywall with large perforations
Since the holes are deep and there are many of them, I wouldn't try to paint them. I'd use a thin-nap (1/4") or foam roller and just paint the surface of the wall. If you keep the roller fairly well wrung out you shouldn't see significant squeeze into the holes. 

You might need to have a scrap of plywood or cardboard on hand to pre-roll, making the paint load more uniform and light. Also, it may take several light coats. 



Oops. I was focusing on the dimples on the horizontal panels and didn't see that the ones on the wall in that photo are different. 

Original answer:

I would use a foam brush and dab all the dimples first, then a thin-nap (1/4" or thinner--maybe a foam roller) to roll the entire wall. Apply several light coats so you aren't squeezing paint out into the dimples. 

This would also be a good use case for a sprayer. Either a rented commercial sprayer or an inexpensive airless should do.