Can someone who sincerely believes when young lose their faith in old age?
Until the last breath anything is possible. But someone who is truly sincere will not meet such an end; the one at risk is the person who only thinks he is sincere while carrying contradictions within.
Many people know the worry of whether their faith will last a lifetime. To be honest, anything is possible until the last breath. No one can sit back and say that a good end is guaranteed. This honesty is part of faith.
Yet there is no reason to fear for someone who is truly sincere. God does not let a person fall in old age who serves Him honestly and with all his heart. The first verse above says that God strengthens the believers with the firm word, in this life and in the next.
When someone abandons faith in old age, the reason is usually not old age itself. Often the sincerity was not as deep as it seemed from outside. Some people consider themselves sincere but carry contradictions within. They believe in God and at the same time cling to something that matters more to them. Such cracks can stay hidden for a long time. Under the pressures of life or in old age they may break open.
So the way forward is to examine oneself honestly while one still can. Where are my contradictions? What really matters most to me? Whoever takes these questions seriously builds their house on firm ground.
Prayer is the other part. The second verse above and the hadith show that even the best of people asked God not to let their hearts go astray. Steadfastness is nothing to be proud of. It is a gift from God that one asks for again and again.