Greetings junkbunny :)
A couple of ways you could look at it :
1. I believe it is not speaking of time at all, but is connecting it as an adjective to the age it is referring to. In other words, it is basically speaking of having either punishment that is pertaining to the age, or life that is pertaining to the age. That age is referring to the Messianic age, the one they were looking for. Because it is not speaking of time, there are other terms that describe our condition, such as putting on immortality. The punishment that is pertaining to the age is nowhere said to never end. It, being corrective in nature, accomplishes its purpose, and thus, is no longer needed.
2. Some believe it is referring to a length of time, described as age-abiding, or age-lasting. In this case, again, the life or punishment is only being described as lasting as long as that age. In this case, the belief is that that age will end, and when it does, that punishment will as well...however, again, the life a person has must continue on because we must put on immortality. The life we have after that age will not be age-abiding life, but simply life outside of the ages, for the ages will end.