We present MANTRA, a framework for approximating the temporal betweenness centrality of all nodes in a temporal graph. Our method can compute probabilistically guaranteed high-quality temporal betweenness estimates (of nodes and temporal edges) under all the feasible temporal path optimalities, presented in the work of Buß et al. (KDD, 2020). We provide a sample-complexity analysis of our method and speed up the temporal betweenness computation using a state-of-the-art progressive sampling approach based on Monte Carlo Empirical Rademacher Averages. Additionally, we provide an efficient sampling algorithm to approximate the temporal diameter, average path length, and other fundamental temporal graph characteristic quantities within a small error $\varepsilon$ with high probability. The running time of such approximation algorithm is $\tilde{\mathcal{O}}(\frac{\log n}{\varepsilon^2}\cdot |\mathcal{E}|)$, where $n$ is the number of nodes and $|\mathcal{E}|$ is the number of temporal edges in the temporal graph. We support our theoretical results with an extensive experimental analysis on several real-world networks and provide empirical evidence that the MANTRA framework improves the current state of the art in speed, sample size, and required space while maintaining high accuracy of the temporal betweenness estimates.