The post-COVID-19 era means that the COVID-19 is basically under control; however, the risk of the pandemic still affects people's work, study, and life, physically and psychologically. In this era, due to the more challenges first-year college students face, more attention should be paid to their mental health. An emerging study demands-resources (SD-R) model can explain the influencing mechanism of college students' mental health. This model suggests that study demands increase the risk of student burnout, which results in mental health problems; meanwhile, study resources reduce student burnout and increase student engagement, thus improving mental health. Based on the SD-R model, this study explores the impacts of time pressure, emotional exhaustion, perceived social support, and student engagement on mental health and provides adequate measures to reduce the risk of mental health problems among first-year students. Time pressure, perceived social support, emotional exhaustion, student engagement, and mental health scales were used to investigate 537 first-year students at three universities in Guangxi, China, of whom 290 (54%) were female, and 247 (46%) were male, and the average age was 18.97 ± 1.01. Results indicated that: (1) Moderate scores on time pressure and emotional exhaustion and slightly-above-the-median scores on perceived social support, student engagement, and mental health were found among first-year students in the post-COVID-19 era. (2) Time pressure had a positive relationship with emotional exhaustion and a negative relationship with mental health. (3) Perceived social support was negatively correlated with emotional exhaustion but positively correlated with student engagement, and thus improved mental health. Results of this study with a sample of first-year college students in China support the hypotheses based on the SD-R model. These findings suggest that increasing perceived social support and student engagement while decreasing time pressure and emotional exhaustion may promote mental health among first-year college students.