We consider a spectrum sharing problem where two users attempt to communicate over N channels. The Primary User (PU) has prioritized transmissions and its occupancy on each channel over time can be modeled as a Markov chain. The Secondary User (SU) needs to determine which channels are free at each time-slot and attempt opportunistic transmissions. The goal of the SU is to maximize its own throughput, while simultaneously minimizing collisions with the PU, and satisfying spectrum access constraints. To solve this problem, we first decouple the multiple-channel problem into N single-channel problems. For each decoupled problem, we prove that there exists an optimal threshold policy that depends on the last observed PU occupancy and the freshness of this occupancy information. Second, we establish the indexability of the decoupled problems by analyzing the structure of the optimal threshold policy. Using this structure, we derive a Whittle index-based scheduling policy that allocates SU transmissions using the Age of Information (AoI) of accessed channels. We also extend our insights to PU occupancy models that are correlated across channels and incorporate learning of unknown Markov transition matrices into our policies. Finally, we provide detailed numerical simulations that demonstrate the performance gains of our approach.