Links in practical systems, such as satellite--terrestrial integrated networks, exhibit distinct delay distributions, intermittent availability, and heterogeneous energy costs. These characteristics pose significant challenges to maintaining timely and energy-efficient status updates. While link availability restricts feasible transmission routes, routing decisions determine the actual delay and energy expenditure. This paper tackles these challenges by jointly optimizing sampling and routing decisions to minimize monotonic, non-linear Age of Information (AoI). The proposed formulation incorporates key system features, including multiple routes with correlated random delays, stochastic link availability, and route-dependent energy consumption. We model the problem as an infinite-horizon Constrained Semi-Markov Decision Process (CSMDP) with a hybrid state--action space and develop an efficient nested algorithm, termed Bisec-\textsc{ReaVI}, to solve this problem. We analyze the structural properties of the solution and reveal a well-defined jointly optimal policy structure: (i) For general monotonic penalty functions, the optimal sampling policy is a piecewise linear waiting policy with at most $N$ breakpoints given $N$ routes; and (ii) under a derived Expected Penalty Ordering condition, the optimal routing policy is a monotonic threshold-based handover policy characterized by at most $\binom{N}{2}$ thresholds. Numerical experiments in a \textit{satellite--terrestrial} integrated routing scenario demonstrate that the proposed scheme efficiently balances energy usage and information freshness, and reveal a counter-intuitive insight: \textit{even routes with higher average delay, higher delay variance or lower availability can still play a critical role in minimizing monotonic functions of AoI}.

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