Time Sensitive Networking (TSN) is fundamental for the low-latency, reliable, and energy-efficient networks that will enable the Industrial Internet of Things (IIoT). Wi-Fi has historically been considered unfit for TSN, as channel contention and collisions prevent deterministic transmission delays. How- ever, this issue can be overcome using Target Wake Time (TWT) to instruct Wi-Fi stations to wake up and transmit in non- overlapped TWT Service Periods (SPs) and sleep in the remaining time. In this paper, we first formulate the TWT Acceptance and Scheduling Problem (TASP), whose objective is to schedule TWT SPs as to maximize traffic throughput and energy efficiency while respecting Age of Information (AoI) constraints. Then, since the TASP is NP-hard, we propose the TASP Efficient Resolver (TASPER), a heuristic strategy to find near-optimal solutions efficiently. Finally, we compare TASPER with several baselines through numerical analysis and simulations, which we performed using a TWT-compatible simulator based on ns-3. We demonstrate that TASPER schedules traffic with up to 21.23% higher priority-weighted admission ratio and saves up to 7.42% energy compared to the ShortesFirst strategy, all while satisfying AoI constraints for 99.5% of transmissions.