First of all I think the question, phrased like this, assumes and generalizes. Yes, some old people are not afraid of dying, but some are just as afraid. Life is precious, and you'd be surprised to know just how deep the human urge for self preservation goes. In fact according to some researches it is equal only to sexual desires, which itself aims at a sort of preservation, the preservation of one's species. Which follows that by preserving your specie you preserve a part of yourself in them. Which brings me to the first point in answering your amazing question.
For those old people who aren't mortified by death, what exactly drives their confidence in the face of death?
1. MAKING PEACE WITH MORTALITY
We all know life is short and no one lives forever. That is the beauty of life. In fact, according to Toni Morrisson, that IS the meaning of life. From childhood we've learnt about this. We've had our whole lives to prepare for it.
Life has its viccisitudes. At a point life's great and everything's rainbows and unicorns, then it isn't. All the ups and downs are there to teach us just how feeble life is. We make friends, we lose friends. We make enemies, we lose enemies. We lose familiy members and loved ones and in the end at one point or another you start to contemplate your own mortality.
It might seem distasteful at first. You might rage against it, like Dylan Thomas said in his poem, but in the end you start to make peace with it. You start to content yourself with thoughts of enjoying the moments. And when you grow old you can look back on those moments you've had and be thankful for them.
2. PRESERVATION
Second, as i mentioned in the introduction, is preservation. As a kid, self preservation is paramount. As we grow old however and start to contemplate our mortality; start to wonder what will become of us in the future years to come, we start to think of ways by which we can preserve our legacies. First among them, like I said, is through our sexual urges; subliminating this into the preservation of the specie via our offsprings.
Much joy can be derived from this at old age, where you feel satisfied by the fact that you have preserved a copy or copies of you on earth to carry on your legacies.
Another way of posthumous preservation might be through deeds. Deeds of great men are honored and remembered. We still talk about and study men who lived thousand years ago. A man or woman who has done deeds s/he feels proud of also tend to feel less fear at the thought of death.
3. RELIGION
Various religions have different beliefs for what haopens after death. That is in fact the basis for most religion. A man or woman who feels his or her life has been lived according to the spiritual tenets of his religion might feel less afraid at the thought of death.
S/he would feel that since their lives on earth had been spent judiciously and piously, there is nothing to fear in the afterlife.
It should be worth noting that some religions don't believe in the afterlife and for these the consequence is neutral..
5. VAPIDITY
Pleasure palls. Life becomes boring. It is true that at the point where you seem to no longer be capable of having a thing you start to value that thing the more, and that is why most of people at their death beds pray for a second chance.
But also most people who have gotten old stay content with the fact that THEY have lived. Death would seem easier for those who have lived and loved to the fullest. Which is why we are always being admonished to not live our lives not for anybody but ourselves, life's too short. YOLO.
4. MORTIFICATION
By mortification I mean genuine mortification/disattachment of the body--which some people do in fact attain. These "mystics" have attained a complete disjunction from eartly things; pleasures, pain, thrills, anything that may have caused them to feverishly cling to life
To them life, the body, isn't the absolute, but the soul. And it is in the preservation of one's soul that true bliss lies. And that can still be achieved even in death.
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So there you have it; depending on your beliefs and dispositions, these are ways by which you can assure you're not afraid when the time comes. Cheers.
Also I'll reccomend reading Ursula le Guinn's take on growing old happily here: https://www.brainpickings.org/2014/10/21/ursula-le-guin-dogs-cats-dancers-beauty/