This work investigated the fault-resilient output-feedback control problem for delayed memristor-based neural networks (MBNNs) subject to actuator faults and external disturbances. To overcome the practical limitation that full-state measurements are often unavailable, an output-feedback controller was designed to rely solely on measurable outputs. By constructing a Lyapunov-Krasovskii functional and employing the Bessel-Legendre inequality together with the generalized reciprocally convex combination inequality, a sufficient condition was established to simultaneously guarantee ${\mathcal{H}}_{\mathrm{\infty }}$ and ${\mathcal{L}}_2-{\mathcal{L}}_{\mathrm{\infty }}$ disturbance attenuation for the considered MBNNs. Decoupling techniques were then introduced to eliminate nonlinear coupling terms, and the controller synthesis was formulated in terms of linear matrix inequalities that are solvable with existing convex optimization tools. Finally, a numerical example was provided to corroborate the reduced conservatism of the proposed analysis condition and the effectiveness of the synthesis method in achieving the desired dual-performance guarantees.