This paper analyzes and optimizes the average Age of Information (AAoI) of Frame Slotted ALOHA with Reservation and Data slots (FSA-RD) in a multi-access network, where multiple users transmit their randomly generated status updates to a common access point in a framed manner. Each frame consists of one reservation slot and several data slots. The reservation slot is further split into several mini-slots. In each reservation slot, users that want to transmit a status update will randomly send short reservation packets in one of the mini-slots to contend for data slots of the current frame. The reservation is successful only if one reservation packet is sent in a mini-slot. The data slots are then allocated to those users that succeed in the reservation slot. In the considered FSA-RD scheme, one user with a status update for transmission, termed active user, may need to perform multiple reservation attempts before successfully delivering it. As such, the number of active user(s) in different frames are dependent and thus the probability of making a successful reservation varies from frame to frame, making the AAoI analysis non-trivial. We manage to derive an analytical expression of AAoI for FSA-RD by characterizing the evolution of the number of active user(s) in each frame as a discrete-time Markov chain. We then consider the FSA-RD scheme with one reservation attempt per status update, termed FSA-RD-One. Thanks to the independent frame behaviors of FSA-RD-One, we attain a closed-form expression for its AAoI, which is further used to find the near-optimal reservation probability. Our analysis reveals the impact of key protocol parameters, such as frame size and reservation probability, on the AAoI. Simulation results validate our analysis and show that the optimized FSA-RD outperforms the optimized slotted ALOHA.