There is a lot of money currently being invested by your friendly silicon valley billionaires in the hope that if we take the right drugs and supplements, which affect the correct genes, that aging will be arrested or even reversed.
So you can just go ahead and say your character got lucky with some genetic mutations which extend his telomeres or something, and you're good.
Doubtlessly, aging is a biological process, and with sufficient knowledge of biology man will one day become functionally immortal as you describe.
There is no real reason to suppose that aging will be easy to fix. This is because its not simply a matter of restoring a lost ability. Bodies have never been immortal.
Once upon a time, our ancestors reproduced by dividing in two. Obviously, if you're a single cell, you have to be immortal in order to reproduce. These immortal cells are called the germ line.
Eventually the single cell evolved to surround itself with other cells, all working together to make sure that germ line cell could reproduce.
Those other cells were expendable. They are body cells. But our gametes (egg, sperm) are part of an immortal cell line (your germ line) that goes back to the first cell that ever was.
Our bodies, however, have never needed to be immortal, as long as the germ line continues on. So even though they all come from immortal germ line cells, as evolution continued body cells quickly lost their ability to be immortal.
Only the simplest animals (some jellyfish for example) are able to fully repair their bodies.
The more complex bodies have become, the more difficult they are to maintain, and evolution has never had a reason to worry about what happens to you after your children have grown into functional adults.
From the perspective of your immortal reproductive cells, you are expendable, like a car. You can maintain your car, replace parts, but eventually you have to get a new car.
That is the way your body is in relation to your germ line. When you have a child your germ line gets a new body and leaves you behind.
Unfortunately this means things look pretty bad for our chances of finding a fountain of youth.
Additionally, it is safe to say that the chances of a combination of mutations resulting in an immortal individual is extremely slim. Its not just a matter of changing a few things, its a matter of adding huge amount of genetic material to accomplish all of the following things:
Repair or replacement of body parts that are not actually made of cells. (parts of teeth, parts of eyes, parts of bones, parts of connective tissue) AND full removal of damaging plaques in the brain and arteries.
-----> this requires a bunch of new genetic processes that no animal has. (note that immortal jellyfish don't have teeth, eyes, bones, or much of a brain)
Drastically increased resistance to cancer, dementia, and other diseases that are associated with aging but not directly caused by it.
------> The longer you live, the more likely you'll eventually get cancer, even if you don't age, your cells are still accumulating damage from free radicals, UV rays and random DNA transcription mistakes. New genetic processes are needed.
Better immune system.
------> Living 1000 years means living through multiple plagues, etc. Without a better immune system that is unlikely. You could say they just got lucky, but if not then they need genetic changes for this too.
------> While not needed to simply live, it would be important for the character to retain his/her youthful ability to learn new ideas and adapt. By the time someone is 28 they are pretty set in their ways, but living 1000 years means they will need to reinvent themselves over and over, and learn new languages and skills multiple times if they are to appear "normal" to the people around them and not get lynched. This will require some genetic changes as well.
The ability to regrow limbs, heal without scarring.
-------> scars are the body's way of just patching over an injury as soon as possible. If you're mortal it doesn't really matter if you accumulate scars and unhealed injurie, but for an immortal the accumulated scars would eventually be debilitating. Fixing this issue would require new DNA dedicated to enabling full healing of any and all injuries that are not immediately fatal, including non-traumatic injuries and internal injuries like heart attacks and strokes.
There are far too many changes needed for this to have happened in one single individual through a lucky combination of mutations. Baring divine or extraterrestrial tampering, there is only one plausible way this could EVER have happened naturally: your character is from an ancient race of individuals that only reproduced when they were old.
Only in this way would evolution be able to act effectively to add the hundreds or thousands of new genes needed to allow for a long lifespan.
Each generation of individuals would have to reproduce ONLY at the last third of their life.
First, selection would favor females with extended reproductive fertility.
Then it would favor those who lived long enough to take care of their children.
Gradually, over many thousands of generations, lifespans would begin to expand to hundreds of years.
Any early breeding would have ruinous consequences for the program. One can imagine the kind of appalling social structures this would necessitate. No sex until you're 600. Early babies must be killed. Etc...
In order to sustain this breeding program in the face of overwhelming competition from faster-breeding normal humans who would overwhelm them and kill them in normal circumstances, they would have to have been insulated since prehistoric times by some kind of natural barrier. Perhaps they lived on a large island or in a hidden valley behind a high mountain range.
Obviously there are no such places around today, but since your character is 1000 years old, you could say they were found by the rest of civilization 1000 years ago, and subsequently destroyed by outsiders, leaving your character as the only remaining immortal.