We consider a star-topology wireless network for status update where a central node collects status data from a large number of distributed machine-type terminals that share a wireless medium. The Age of Information (AoI) minimization scheduling problem is formulated by the restless multi-armed bandit. A widely-proven near-optimal solution, i.e., the Whittle's index, is derived in closed-form and the corresponding indexability is established. The index is then generalized to incorporate stochastic, periodic packet arrivals and unreliable channels. Inspired by the index scheduling policies which achieve near-optimal AoI but require heavy signaling overhead, a contention-based random access scheme, namely Index-Prioritized Random Access (IPRA), is further proposed. Based on IPRA, terminals that are not urgent to update, indicated by their indices, are barred access to the wireless medium, thus improving the access timeliness. A computer-based simulation shows that IPRA's performance is close to the optimal AoI in this setting and outperforms standard random access schemes. Also, for applications with hard AoI deadlines, we provide reliable deadline guarantee analysis. Closed-form achievable AoI stationary distributions under Bernoulli packet arrivals are derived such that AoI deadline with high reliability can be ensured by calculating the maximum number of supportable terminals and allocating system resources proportionally.