well initially it's hard to determine the fate of captain rex since he is one of the supporting characters in the clone wars tv series and since the series is airing in its early episodes its really hard to predict whether or not he will get killed or not, but as the series will reach its conclusion only then we will know what will happen to him.
The answer is yes. All the clones die. Due to there age acceleration they age twice as fast as normal humans. Therefore he died after the clone wars in old age.