I have had this problem with various items, including recently a pair of binoculars. I cannot remember what I used to get rid of the stickiness as I think I tried everything evil in the workshop, but it was some solvent- possibly methylated spirits- and the decomposing layer of the 'rubber' was removed. The stickiness soon returns though. It does not seem to be caused by UV as the binoculars have spent their life in a case from new.
I read on another forum a while ago that there is a rubber paint available and this can be applied in several coats with a brush to replace the original coating after it has been removed entirely.
BT