[Gothmog]

Just look at Elros! He chose to become mortal, took Tar-Minyatur and lived for 500 years.

And why should Arwen die at once? She was old, that's true but her body wasn't old like a human body would be after that long. Apparently elves can keep their bodies young, hence their timeless looks. She shouldn't die of old age until her mortal body has grown old. You or me wouldn't die even if we lived for 500 years if our body could find a way of preserve itself, or even better renew itself. A person isn't killed by age itself, but by the effect it has on the body.

I don't know how giving up mortality works, but I can't see a reason why Arwen should die the moment she declares herself mortal.