Five phenolic Schiff bases (7–11) incorporating a fragment of methanesulfonamide were synthesized and evaluated for their efficacy as antitumor agents. Compounds 7 and 8 demonstrated the most potent antitumor action, with a positive cytotoxic effect (PCE) of 54/59 and 59/59 and a mean growth percentage (MG%) of 67.3% and 19.5%, respectively, compared with imatinib (PCE = 20/59 and MG% = 92.6%). The PCE values for derivatives 9–11 were 3/59, 4/59, and 4/59, respectively, indicating poor antitumor effect. Compound 8 exhibited the most significant efficacy, suppressing cell proliferation by an average of 50% at a dosage of 0.501 µM, in comparison with the reference drugs sorafenib (2.33 µM), gefitinib (2.10 µM), erlotinib (7.68 µM), and celecoxib (17.5 µM). Compounds 7 and 8 had substantial inhibitory effects on the human epidermal growth factor receptor 2 (HER2), with IC50 values of 0.183 μM and 0.464 μM, respectively. Furthermore, they exhibited significant inhibition of the epidermal growth factor receptor (EGFR), with IC50 values of 0.752 μM and 0.166 μM, respectively. Compound 8 exhibited the highest COX-2 inhibition (IC50 = 12.76 μM). We performed molecular docking dynamic experiments to examine the precise interaction and structural prerequisites for the anticancer activity of derivatives 7 and 8 by targeting EGFR and HER2.