How does a sperm cell die?
So most cells require growth factors and survival factors to stay alive in an organism. Cells which stop receiving survival factors, such as sperm, undergo a process of programmed cell death (PCD) called apoptosis. Some cells will also undergo necrosis, or autophagy, among other types of PCD. Lachaud et al. performed a study in 2004 on the contribution of apoptosis and necrosis in ejaculated sperm. They make note in the intro that from other literature that we see some hallmarks of apoptosis like externalized phosphatidylserine, fragmented DNA, etc. despite inactivated nuclei and a small cytoplasm. Their data notes that ejaculate sperm cell death mainly occurs from necrosis, however, because sperm themselves lack the ability to initiate apoptosis. However, in the testes, there's data to note that apoptosis plays a role, and the presence of apoptotic spermatozoa in the ejactulate may be result of the inititation of apoptosis by factors which still inside the body. Apoptosis of spermatozoa in the testes is currently under investigation in male infertility, and damage in cryopreservation.

Just to give you an idea of the difference, necrosis is a much messier outcome derived from physical factors, or those factors external to the cell. Apoptosis is organized, controlled, and modulated by cellular signals.