We propose two Euler-Maruyama (EM) type numerical schemes in order to approximate the invariant measure of a stochastic differential equation (SDE) driven by an $α$-stable Lévy process ($1<α<2$): an approximation scheme with the $α$-stable distributed noise and a further scheme with Pareto-distributed noise. Using a discrete version of Duhamel's principle and Bismut's formula in Malliavin calculus, we prove that the error bounds in Wasserstein-$1$ distance are in the order of $η^{1-ε}$ and $η^{\frac2α-1}$, respectively, where $ε\in (0,1)$ is arbitrary and $η$ is the step size of the approximation schemes. For the Pareto-driven scheme, an explicit calculation for Ornstein--Uhlenbeck $α$-stable process shows that the rate $η^{\frac2α-1}$ cannot be improved.