Hello kr3t3n,
I haven't grown potatoes hydroponically yet, but do have plans to grow them using an
aeroponic system. My plans include building a insulated box with a removable side. Insulated to help keep the roots cool. A removable side in order to be able to access the root system and be able to harvest the potatoes while the plants are still growing.
I wouldn't use the netting inside, otherwise the root system will become attached and intertwined in the netting. You won't be able to harvest the potatoes without killing the whole plant. If you tried to just cut the potatoes off and leave the dead roots attached to the netting, you'll provide food for pathogens and cause disease. I also wouldn't use a fogger for a few reasons. I would use mister/sprinkler heads at the top and let the water/
nutrients rain down over the root system. You can use either a high pressure or low pressure system, but the low pressure system would be much simpler and cheaper to build.
If you want to use a 55 gallon barrel, I would create a door/opening on one side large enough to access all the roots inside. Make sure it's waterproof when closed. Then insulate the barrel to keep the root zone cool. Make small openings in the top of the barrel to place the mister/sprinkler heads into for easy access and maintenance, and so the water/
nutrient will rain down over the entire root system. Then assuming it's a recirculating system raise it up high enough so it's above the reservoir, and create water return line from the bottom of the barrel back to the reservoir.