This work concerns the Ornstein-Uhlenbeck type process associated to a positive self-similar Markov process $(X(t))_{t\geq 0}$ which drifts to $\infty$, namely $U(t):= {\rm e}^{-t}X({\rm e}^t-1)$. We point out that $U$ is always a (topologically) recurrent Markov process and identify its invariant measure in terms of the law of the exponential functional $\hat I := \int_0^\infty \exp(\hatξ_s) {\rm d}s$, where $\hatξ$ is the dual of the real-valued Lévy process $ξ$ related to $X$ by the Lamperti transformation. This invariant measure is infinite (i.e. $U$ is null-recurrent) if and only if $ξ_1\not \in L^1(\mathbb{P})$. In that case, we determine the family of Lévy processes $ξ$ for which $U$ fulfills the conclusions of the Darling-Kac theorem. Our approach relies crucially on another generalized Ornstein-Uhlenbeck process that can be associated to the Lévy process $ξ$, namely $V(t) := \exp(ξ_t)\left(\int_0^t \exp(-ξ_s){\rm d}s +V(0)\right)$, and properties of time-substitutions based on additive functionals.