I finally got this to work. The trick is:
The DrawIcon() function (from WinAPI.Windows) doesn't work with Bitmaps of the type FMX.Types.TBitmap which the TImages now have under Firemonkey. So I had to create a temporary Bitmap of the type Vcl.Graphics.TBitmap on which I can draw the cursor icon. I than create a temporary MemoryStream and use the SaveToStream method of my temporary Bitmap to write it to that MemoryStream from which I can load it to the TImage on my form (Image1.Bitmap.LoadFromStream(TmpMemoryStream)).