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.
I ran into the same problem when I got to the Rear Retaining Rail. It appears that mine has some type of Yellow Adhesive holding the Retaining Rail to the Rear Bow. I pried gently with a Screwdriver to try to remove it, but to no avail. I also removed the Main/Large Door Speaker Covers to replace the speakers, and decided to bring them inside to clean them as well (they are filthy from dirty shoes hitting them over the years.
I may try a Heat Gun tomorrow on the rear retaining rail to see if I can get the adhesive to let go. Are all of these glued except the one in the video? Has anyone else had this same experience? My Headliner is FILTHY and Water Stained!