I'm asking this since Brom was revealed to be a rider. His dragon, the original Saphira, died about 100 years before the story line started. He was still alive, but he bore the appearance of an older man. So was his aging process just slowed, or was he still immortal?
It depends on how long they were bonded together.
Christopher Paolini answered this very question on his recent Reddit AMA.
Is a rider, who lost their dragon still immortal, or do they just possess extremely long life?
If the Rider is an elf, then yes. If a human, then it depends on how long the human and dragon were bonded together (the longer they were, the more elf-like the human becomes). Brom was only bonded to his Saphira for a few years, and yet he still lived far longer than a normal human.
I think they are not immortal(they can die) but only if they have a mortal wound. otherwise I think they would just keep going. As we can see, Brom died due to his wounds. but even if he was worn down, he still had a lot of energy and his appearance did not hint that he was older than a normal old human. So I guess the time depends on how well he takes care of himself. so no he is not immortal. but he is not naturally mortal either.