You do NOT need damage to be set to full for this achievement, visual only will do.
The first Rally event in the Dirt Tour with the 60's class available is in Season 1 - Rockstar Transworld Clash (Top segment) 'Tsavo Classic Shield'.
You dont have to do this specific event but this is the first event in the Dirt Tour that the Mini is available for use, select the event and select the class 60's and select any of the Mini Cooper S models (Team choice doesnt matter)
Now you need to not only complete the rally without damaging but you must also WIN so i suggest you race on the lowest difficulty (1), take it easy, follow the suggested braking line and dont forget to make use of your flashbacks.
It's ok to go off the main track a bit as long as the car doesnt hit anything and the small fences (Thin poles and tape) are OK to hit too.
The achievement will pop at the loading screen, the event is a 2 stage event so if you happen to mess up the first go you have another go without wasting your time, good luck.