Weighted extropy has recently emerged as a flexible information measure for quantifying uncertainty, with particular relevance to order statistics. In this paper, we introduce and study a weighted cumulative analogue of extropy, extending the framework of weighted cumulative residual and cumulative past entropies to extreme order statistics. Specifically, we define the general weighted cumulative residual extropy (GWCREx) for the smallest order statistic and the general weighted cumulative past extropy (GWCPEx) for the largest order statistic, along with their dynamic versions. We show that these weighted measures and their dynamic counterparts uniquely characterize the underlying distribution. Moreover, we establish new characterization results for two widely used reliability models: the generalized Pareto distribution and the power distribution. The proposed framework provides a unified information-theoretic tool for analysing extreme lifetimes in reliability engineering and survival analysis.