In this paper, we study a fresh data acquisition problem to acquire fresh data and optimize the age-related performance when strategic data sources have private market information. We consider an information update system in which a destination acquires, and pays for, fresh data updates from a source. The destination incurs an age-related cost, modeled as a general increasing function of the age-of-information (AoI). The source is strategic and incurs a sampling cost, which is its private information and may not be truthfully reported to the destination. To this end, we design an optimal (economic) mechanism for timely information acquisition by generalizing Myerson's seminal work. The goal is to minimize the sum of the destination's age-related cost and its payment to the source, while ensuring that the source truthfully reports its private information and will voluntarily participate in the mechanism. Our results show that, under some distributions of the source's cost, our proposed optimal mechanism can lead to an unbounded benefit, compared against a benchmark that naively trusts the source's report and thus incentivizes its maximal over-reporting.