If you get some cooking oil into your eye while cooking you should flush the eye with cool water for 15 to 20 minutes to help flush the cooking oil out and help cool down your eyes.
Then after you cool down your eye use a washcloth with water to clean the rest of the oil out around your eye.
Then use some moisturizing eye drops in your eyes once you flush the cooking oil out and have cooled down your eyes.
Your eye might be red for a little and still may have some stinging and burning for a few days but it should be okay and if you're worried or it doesn't seem to get better then going to your eye doctor or urgent care would be a good idea.