The sperm cells themselves are very short-lived. However, they do originate from spermatogonial stem cells that keep on replicating theough all your life.

In a cartoon, these stem cells divide into two cells. One of these daughter cells (cell A) keeps being a stem cell and the other not (cell B), begining a chain of divisions into further daughter cells that end up being a bunch of sperm.

New mutations appearing on cell B will only be found in one "round" of sperm. Mutations appearing in cell A are there to stay in all future rounds.