Identifying coping styles in cats may lead to improved health and welfare. The aims of this study were to (1) identify individual differences in response to acute confinement, and (2) to assess the predictability of guardian-rated personality traits on behavior. Adult cats (n = 55) were singly housed in enriched cages and behavioral observations were recorded for three days. On day 3, familiar and unfamiliar person approach tests were conducted. Fecal glucocorticoid metabolites (FGM) were quantified from voided samples. A questionnaire assessing personality traits and sickness behaviors was completed by each guardian. Analysis identified two clusters—cats in Cluster 1 (n = 22) were described as shy, calm, mellow, and timid; cats in Cluster 2 (n = 33) were described as active, playful, curious, and easygoing. Multilevel mixed-effects GLM revealed significant differences between the clusters including food intake (C1 > C2, p < 0.0001), affiliative/maintenance behaviors (C2 > C1, p < 0.0001), vocalization (C2 > C1, p < 0.0001), hide (C1 > C2, p < 0.0001), perch (C2 > C1, p < 0.0001), and latency to approach a familiar (C1 > C2, p < 0.0001) and unfamiliar (C1 > C2, p = 0.013) person. No statistically significant differences in FGM concentrations were identified (cluster p = 0.28; day p = 0.16, interaction p = 0.26). Guardian-rated personality traits agreed with the response of the cats when confined to a cage, suggesting that domestic cats have different coping styles. Identifying individual differences in response to stressful events or environments may provide caretakers with important information leading to improved welfare.