Rearchitect the app to point it at the new site instead of reddit.
Model it after Wikipedia's monetization. Never advertise, maybe some monetization through reddit gold style status things you can buy but mostly keep the lights on by begging for money a couple times a year.
Make it nonprofit like Wikimedia foundation.
Keep reddit's structure of subdirectories. Replace the UI with what the app looks like.
Keep costs low initially by having videos and images hosted off site, text only.
Hard to say if this would make enough money to keep the lights on but would be interesting to see it attempted.