I would have much preferred to have been able to take the headliner out altogether. Maybe yours will come off like the video above...
Mine is a 1999 car and removal was fine and as per video above until I got to the rear of the headliner, where in my car it seemed wrapped or glued to the rear of the soft top frame so I went no further.
I cleaned it while attached at the rear but otherwise it was hanging.
So, I put down sheets to protect the interior.
My headliner is light grey. If I was black I wonder if water marks would come up easily?
Panel by panel so as to avoid water marks, I sprayed Simoniz foam action upholstery cleaner, waited a few minutes, used a damp cloth and a touch of water and very gently scrubbed. For anything a little stubborn I used a sponge and agitated it with the grain of the headliner fabric. As best I could I added a little water with a sponge to dilute the foam cleaner. Using micro fibre cloths and light coloured towels I dabbed to soak out residue. If headliner was black I would have used dark towels in case of shedding. I then left the headliner overnight and in the morning it was clean and undamaged. It is still spotless two and a half years later.
Maybe others can advise you better. Be gentle.