Let $X$ and $Y$ be superreflexive complex Banach spaces and let $\Cal{L}(X)$ and $\Cal{L}(Y)$ be the Banach algebras of all bounded linear operators on $X$ and $Y$, respectively. We describe a linear map $\phi:\Cal{L}(X)\to\Cal{L}(Y)$ that almost preserves the approximate point spectrum or the surjectivity spectrum. Furthermore, in the case where $X=Y$ is a separable complex Hilbert space, we show that such a map is a small perturbation of an automorphism or an anti-automorphism.