Anti-Zeno enhancement of quantum heat machines has been established for working fluids driven by an external classical field, but whether it can enhance extractable-work accumulation in an autonomously evolving quantized piston remains an important question. Here we study an autonomous thermal machine in which a two-level working fluid is coupled to a quantized harmonic piston and to two spectrally separated reservoirs, without external modulation. We derive a finite-time master equation for the resulting carrier and sideband channels, whose rates interpolate between the Zeno, anti-Zeno, and Markovian regimes as the coupling time is varied. For an initially coherent piston, the machine charges the piston and generates more ergotropy than the Markovian reference in the anti-Zeno window. Reversing the same retained channels turns the device into a finite-resource refrigerator with enhanced cooling. The anti-Zeno effect therefore increases the rate of operation while leaving the underlying carrier and sideband energy ratios unchanged. These channel ratios remain consistent with the usual Carnot bounds over the operating regimes considered here. Our results establish finite-time reservoir sampling as a mechanism for enhancing autonomous thermal-machine operation with a quantized piston.