Altermagnets combine compensated magnetic order with unconventional symmetry-dependent responses, offering a promising platform for spintronic applications. Here, we show that a voltage-controlled magnetic-anisotropy gradient drives altermagnetic (ATM) skyrmions in a rectilinear, Hall-free motion in the absence of thermal fluctuations, owing to their strongly compensated gyrotropic response. Thermal magnons qualitatively modify this behavior by increasing the longitudinal drag through magnon–skyrmion scattering and generating a transverse reaction force through handedness-dependent skew scattering. Owing to the anisotropic altermagnetic magnon band structure, the relative transport weights of the two magnon handednesses are interchanged between propagation along the $x$ and $y$ directions, resulting in opposite signs of the Hall angle for driving along the $x$ and $y$ directions. By contrast, along the high-symmetry direction, the two magnon handednesses remain degenerate and their transverse contributions cancel, preserving Hall-free motion even at finite temperature. We thus uncover a thermally emergent anisotropic skyrmion Hall effect whose direction-dependent magnitude and sign originate from the intrinsic symmetry-dependent magnon spectrum, making it a generic finite-temperature dynamical feature of ATM skyrmions. Our results establish a low-power route toward electrically controlled and thermally tunable ATM skyrmion transport.