We derive a closed-form approximation of the stationary distribution of the Age of Information (AoI) of the semi-persistent scheduling (SPS) protocol which is a core part of NR-V2X, an important standard for vehicular communications. While prior works have studied the average AoI under similar assumptions, in this work we provide a full statistical characterization of the AoI by deriving an approximation of its probability mass function. As result, besides the average AoI, we are able to evaluate the age-violation probability, which is of particular relevance for safety-critical applications in vehicular domains, where the priority is to ensure that the AoI does not exceed a predefined threshold during system operation. The study reveals complementary behavior of the age-violation probability compared to the average AoI and highlights the role of the duration of the reservation as a key parameter in the SPS protocol. We use this to demonstrate how this crucial parameter should be tuned according to the performance requirements of the application.