How can I paint 1000 wooden balls that are 1" each?
Painting by immersion would be fairly easy, but drying would be harder.  Industrially I suspect they'd be kept moving with warm air.

They'd be too expensive for this quantity, and come in sets of too many colours, but some wooden balls are available dyed.  If the ones you can get are pale wood, this may be an option, and drying would be more even than with paint.  You'd soak them a bit then dry them in a tray shaking frequently (or rearranging frequently if you use the support I suggest for spraying.

They could be sprayed.  It would be worth getting a proper sprayer and I'm more familiar with rattle-cans, but the key common factor is that however you spray you do lots of light coats drying in between. You'd rotate between coats so coverage was even. Reusable plastic egg trays are probably the best support, unless you get through huge amounts of eggs packed in plastic trays.  This would take quite a lot of elapsed time, but not much active time - you'd spray a coat in a few minutes even over hundred of balls, then do something else while it dries.

But while looking online I found that 2&nbsp;cm painted balls are available wholesale.  Given your quantities you may be able to find them ready-made after all.