COVID-19, which is caused by Severe Acute Respiratory Syndrome Corona Virus 2 (SARS-CoV-2), has resulted in devastating morbidity and mortality worldwide due to lethal pneumonia and respiratory distress. In addition, the central nervous system (CNS) is well documented to be a target of SARS-CoV-2, and studies detected SARS-CoV-2 in the brain and the cerebrospinal fluid of COVID-19 patients. The blood–brain barrier (BBB) was suggested to be the major route of SARS-CoV-2 infection of the brain. Functionally, the BBB is created by an interactome between endothelial cells, pericytes, astrocytes, microglia, and neurons, which form the neurovascular units (NVU). However, at present, the interactions of SARS-CoV-2 with the NVU and the outcomes of this process are largely unknown. Moreover, age was described as one of the most prominent risk factors for hospitalization and deaths, along with other comorbidities such as diabetes and co-infections. This review will discuss the impact of SARS-CoV-2 on the NVU, the expression profile of SARS-CoV-2 receptors in the different cell types of the CNS and the possible role of aging in the neurological outcomes of COVID-19. A special emphasis will be placed on mitochondrial functions because dysfunctional mitochondria are also a strong inducer of inflammatory reactions and the “cytokine storm” associated with SARS-CoV-2 infection. Finally, we will discuss possible drug therapies to treat neural endothelial function in aged patients, and, thus, alleviate the neurological symptoms associated with COVID-19.