Nay on the fuel additives, although a bottle of techron probably wouldn't hurt. Your 2013 R58 Cooper S should idle fine with regular or premium gas, unless you have a bad tank of gas....

How many miles on the car?  I presume there are no check engine lights?

I would check the conditions of the spark plugs - maybe they will tell you something.

Failing that, there may be a fuel issue at idle (fuel pressure / injector clog)?  Carbon buildup at the valves or on the pistons? Small vacuum leak?  defective turbo bypass valve?  Could be anything....

I recently purchased a new to me, 2013 Cooper S Roadster. Overall, the cars runs great except it idles a bit rough. I've learned this car requires premium fuel and i'm wondering if the idle may have anything to do with the car not having premium fuel in it? For all i know the previous owner (s) may have never used premium fuel.

In my limited research i've read Yea and Nay with regards to adding fuel additives. Need some advice. TY