The sad thing about old school fancy stuff like those wooden cups is they are inherently out dated and not practical. I'm sure you like your cup but if it tastes like greasy old stank oil then the fault obviously lies in the base design. (hence us developing glass, ceramic and steel cups)
Just like animal horn cups, pewter mugs and even those funny rubber foldable cups. They all taste gross. Added with the fact the wooden cups need oil to seal the grain to stop mold growing is just another reason why they taste like 'shoot'. Linseed oil stinks, all my tools stink of it and it makes me feel sick lol.
(I must be getting tired of sealing wood. I hate the smell of bees wax, I hate the smell of linseed oil, teak oil, tung oil etc. All of them remind me of my old work shop which was like the size of a concrete caravan bathroom, tiny and suffocatingly riddled with the stench of oil and wax.)
Regarding getting your cup usable, I can only assume an alchohol fluid gently applied all over and wiped out, repeating that process over a few days to take out the oily coating. After that? no idea.. try parrafin wax? (that's just normal white candle wax)