I am an American and this would be the most common way of saying this. Particularly given the Russian sentence, it is the most correct. There is a minor difference in meaning, "still not old" puts the emphasis on still. Not old yet puts the emphasis on not old. For example if you define old as 70, still not old might mean he is 68 or that he is 72 but does not act his age. Not old yet would mean that he cannot be more than 69.
On the mobile version, where one must choose tiles of words, I had this Russian phrase - "Мой отец ещё не старый" - and the English tiles to choose from were:
I chose the first 6 tiles. I was counted wrong because I did not say, "My daddy is not too old yet."
I take 2 issues with this and I ask if you agree, @kundoo, and will take your advice - (1) отец is father, not daddy, and (2) "too" is not part of this translation, unless there is a set expression or nuance that I am missing? Let me know your thoughts on whether I should report this next time I run across the exercise. Thanks in advance :)