The tradeoff relation between speed and cost is a central issue in designing fast and efficient information processing devices. We derive an achievable bound on thermodynamic cost for obtaining information through finite-time (non-quasi-static) measurements. Our proof is based on optimal transport theory, which enables us to identify the explicit protocol to achieve the obtained bound. Moreover, we demonstrate that the optimal protocol can be approximately implemented by an experimentally feasible setup with quantum dots. Our results would lead to design principles of high-speed and low-energy-cost information processing. Published by the American Physical Society 2024