A +10% Curse item can be found in the first level (Mad Forest). It looks like a purple skull on the map. I was trying to beat 30 minutes when I randomly decided to pick this item up 29 minutes in, just for fun, and I unlocked this achievement even though I only had 10% curse for around 40 seconds.
It seems you only need to survive 20 minutes and then become cursed, rather than survive 20 minutes with a curse on you the whole time.
You can reliably find this item after finding the map upgrade in the 3rd level (Dairy Plant).