Sarcopenia is intricately related to aging associated diseases, such as neuropsychiatric disorders, oral status, and chronic diseases. Dementia and depression are interconnected and also related to sarcopenia. The preliminary shift from robust to sarcopenia (i.e., pre-sarcopenia) is an important albeit underdiscussed stage and is the focus of this study. Identifying factors associated with pre-sarcopenia may lead to sarcopenia prevention. To separately examine the effects of dementia and depression on pre-sarcopenia/sarcopenia, we conducted multiple analyses. This cross-sectional study used health checkup data from a rural Japanese island. The participants were aged 60 years and above, and the data included muscle mass, gait speed, handgrip strength, oral status (teeth and denture), chronic diseases (e.g., hypertension), dementia (cognitive assessment for dementia, iPad Version), and depression (self-rating depression scale). A total of 753 older adult participants were divided into the sarcopenia (n = 30), pre-sarcopenia (n = 125), and robust (n = 598) groups. An ordered logit regression analysis indicated that age and depression were positively correlated with sarcopenia, while hypertension was negatively associated with it. A multiple logistic regression analysis between the robust and pre-sarcopenia groups showed significant associations between the same three variables. Depression was associated with pre-sarcopenia, but not dementia. There was also a significant association between hypertension and pre-sarcopenia. Further research is needed to reveal whether the management of these factors can prevent sarcopenia.