Graph Neural Networks (GNNs) demonstrate superior performance in various graph learning tasks, yet their wider real-world application is hindered by the computational overhead when applied to large-scale graphs. To address the issue, the Graph Lottery Hypothesis (GLT) has been proposed, advocating the identification of subgraphs and subnetworks, \textit{i.e.}, winning tickets, without compromising performance. The effectiveness of current GLT methods largely stems from the use of iterative magnitude pruning (IMP), which offers higher stability and better performance than one-shot pruning. However, identifying GLTs is highly computationally expensive, due to the iterative pruning and retraining required by IMP. In this paper, we reevaluate the correlation between one-shot pruning and IMP: while one-shot tickets are suboptimal compared to IMP, they offer a \textit{fast track} to tickets with a stronger performance. We introduce a one-shot pruning and denoising framework to validate the efficacy of the \textit{fast track}. Compared to current IMP-based GLT methods, our framework achieves a double-win situation of graph lottery tickets with \textbf{higher sparsity} and \textbf{faster speeds}. Through extensive experiments across 4 backbones and 6 datasets, our method demonstrates $1.32\% - 45.62\%$ improvement in weight sparsity and a $7.49\% - 22.71\%$ increase in graph sparsity, along with a $1.7-44 \times$ speedup over IMP-based methods and $95.3\%-98.6\%$ MAC savings.