In this paper, we study age of information (AoI) optimization for status updating in an integrated sensing and communication (ISAC) system. We consider a discrete-time architecture in which a base station interacts with a physical environment and a remote monitor, and at each time slot can operate in one of three modes: sensing, communication, or joint sensing and communication. Each mode is unreliable and incurs a different operational cost. The objective is to minimize a discounted infinite-horizon cost that combines the AoI at the monitor with action-dependent sensing and communication costs. For the single source scenario, we formulate the problem as a Markov decision process with a two-dimensional AoI state and prove that the optimal stationary policy admits an ordered threshold structure in the AoI state space. Since the AoI evolves over an infinite space, we truncate the state space to reduce complexity and rigorously bound the resulting error. The analysis analytically determines the truncation size needed to keep the error below a given threshold. For the multi-source scenario, we formulate the scheduling problem as a restless multi-armed bandit. We develop both a Whittle index policy and an approximate Whittle index policy for scheduling under two different regimes, one where indexability is guaranteed, and one where it is not. Numerical results illustrate the structure of the optimal policy in the single-source case and show that the proposed approximate Whittle index policy performs comparably to the Whittle index policy in the indexable regime, while remaining effective beyond it.