I'm arriving in Kumning by train at 6:40 in the morning and then taking a flight to Guilin at 18:20.

Yes, you can make it. The best way it to hire a car to the scenic area directly and then back to the airport. Upon arrival at the train station, there are a lot of such private cars. The fare is about CNY500 – 600 a day.

You don’t need to store your luggage, just carry it on the car. When you get to the spot, leave the luggage in the car and ask the car driver waiting for you.