Well, it's certainly possible. However, I'm quite sure I'm like most people. Once I solve one problem, another crops up. I'm never without anything to work on.
Is it possible for someone to solve all their problems in their lifetime?
It is possible; it is also dependent upon what one considers a problem and what is merely a situation. We manage what we can control, we deal with what we cannot.
Life might be pretty mundane without challenges.
I think death pretty much solves any problem one may have had. At the very least, it won't matter any more.
With that said if you talk to people who have come close to dying they will often tell you the time and energy spent on many of the things they thought problems were just a waste of time and energy, because they really didn't matter.
If you consider them challenges rather than problems, you can usually solve them easier. Like everyone else said, once you solve one, another pops up.