The problem of anomaly localization in a resource-constrained cyber system is considered. Each anomalous component of the system incurs a cost per unit time until its anomaly is identified and fixed. Different anomalous components may incur different costs depending on their criticality to the system. Due to resource constraints, only one component can be probed at each given time. The observations from a probed component are realizations drawn from two different distributions depending on whether the component is normal or anomalous. The objective is a probing strategy that minimizes the total expected cost, incurred by all the components during the detection process, under reliability constraints. We consider both independent and exclusive models. In the former, each component can be abnormal with a certain probability independent of other components. In the latter, one and only one component is abnormal. We develop optimal simple index policies under both models. The proposed index policies apply to a more general case where a subset (more than one) of the components can be probed simultaneously and have strong performance as demonstrated by simulation examples. The problem under study also finds applications in spectrum scanning in cognitive radio networks and event detection in sensor networks.