with the perch you will need at least 6-8 feet deep they prefer deeper water.
If you start with 100 perch fry in 500 gallons you should be OK. It takes a while for the bacteria to build up. Get your water temperature up to 60 F and the bacteria count will go up a little quicker. Be ready to exchange same temperature water if your ammonia level goes over .5 ppm for more than a day.
Perch will do fine in 4 to 5 feet of water. Just remember they prefer darker water so be sure to shade where you can.
Tell me about your media in the plant beds and I can give you a figure on the amount of plant space needed.
Growing potatoes in aquaponics is fun. I just use a bag and pull it up as the plant grows and I continue to add soil. I use a dripper system that provides just enough to keep the soil moist.