Cognitive impairment is common in Post-COVID-19 Condition (PCC), yet full neuropsychological testing remains resource-intensive. Because eye movements are known to be altered in certain cognitive disorders, Eye-Tracking (ET) offers a fast, non-invasive complementary approach for large-scale screening. This study aimed to predict neuropsychological test scores of participants with PCC from ET metrics using machine and deep learning models. ET data was collected from 172 participants performing a battery of visual tasks designed to elicit smooth pursuit and fixational eye movements, as well as pupil responses to light. Cognitive performance was assessed through established neuropsychological tests. We applied regression and classification models (e.g., Random Forest, XGBoost, and deep neural networks) to predict neuropsychological performance. Models were trained using ET data alone and in combination with the Montreal Cognitive Assessment (MoCA) scores, a widely used neuropsychological test for global cognitive screening. Although predicting individual test scores was challenging, combining them into a global composite measure improved performance. Model sensitivity and specificity reached 88% and 34% using ET data alone, and 87% and 60% when integrating ET with MoCA. This last trained model outperformed the conventional MoCA, highlighting the potential of ET as a rapid screening support tool for cognitive assessment.