Alzheimer’s disease (AD) is recognized as the leading cause of dementia, imposing a significant economic toll on society. Despite the emergence of novel therapeutic approaches for AD, their efficacy and safety mandates further validation through rigorous clinical trials. In this context, hypertension (HTN) has garnered considerable attention as an amendable risk factor for AD. Research indicates that hypertension during midlife is associated with an elevated risk of AD in later years, influencing both the onset and progression of the disease. Nevertheless, the relationship between AD and hypertension in the later stages of life remains a subject of debate. Moreover, the consequences of blood pressure reduction on cognitive function, along with the optimal pharmacological interventions and therapeutic thresholds for hypertension, have emerged as pivotal areas of inquiry. This review synthesizes findings on epidemiology, neuroimaging, and biomarkers, and the effects of antihypertensive medications to elucidate the link between hypertension and cognitive performance. We particularly investigate how hypertension and AD are related by plasma sulfide dysregulation, offering possible indicators for future diagnosis and therapy.