This paper considered a two-stage fourth-order modified explicit Euler/Crank-Nicolson numerical method for solving a time-variable fractional transport equation subject to suitable initial and boundary conditions. Both stability and error estimates of the new approach were deeply analyzed in the $L^{\mathrm{\infty }}(0,T;L^2)$-norm. The analysis suggests that the proposed technique is unconditionally stable and converges with order $O(max\{k^{2-\overline{\beta }},k^{\frac{3}{2}}\}+h^4)$, where $\overline{\beta }$ is a positive constant satisfying $0<\overline{\beta }<1$, $h$ and $k$ are the space step and time step, respectively. This result indicates that the two-stage fourth-order formulation is fast and efficient. Numerical experiments were performed to verify the unconditional stability and convergence rate of the developed algorithm.