Given a graph G, a budget k and a misinformation seed set S, Influence Minimization (IMIN) via node blocking aims to find a set of k nodes to be blocked such that the expected spread of S is minimized. This problem finds important applications in suppressing the spread of misinformation and has been extensively studied in the literature. However, existing solutions for IMIN still incur significant computation overhead, especially when k becomes large. In addition, there is still no approximation solution with non-trivial theoretical guarantee for IMIN via node blocking prior to our work. In this paper, we conduct the first attempt to propose algorithms that yield data-dependent approximation guarantees. Based on the Sandwich framework, we first develop submodular and monotonic lower and upper bounds for our non-submodular objective function and prove the computation of proposed bounds is \#P-hard. In addition, two advanced sampling methods are proposed to estimate the value of bounding functions. Moreover, we develop two novel martingale-based concentration bounds to reduce the sample complexity and design two non-trivial algorithms that provide (1-1/e-ε)-approximate solutions to our bounding functions. Comprehensive experiments on 9 real-world datasets are conducted to validate the efficiency and effectiveness of the proposed techniques. Compared with the state-of-the-art methods, our solutions can achieve up to two orders of magnitude speedup and provide theoretical guarantees for the quality of returned results.