The operation of critical infrastructures such as the electrical power grid, cellphone towers, and financial institutions relies on precise timing provided by stationary GPS receivers. These GPS devices are vulnerable to a type of spoofing called Time Synchronization Attack (TSA), whose objective is to maliciously alter the timing provided by the GPS receiver. The objective of this paper is to design a tuning-free, low memory robust estimator to mitigate such spoofing attacks. The contribution is that the proposed method dispenses with several limitations found in the existing state-of-the-art methods in the literature that require parameter tuning, availability of the statistical distributions of noise, real-time optimization, or heavy computations. Specifically, we (i) utilize an observer design for linear systems under unknown inputs, (ii) adjust it to include a state-correction algorithm, (iii) design a realistic experimental setup with real GPS data and sensible spoofing attacks, and (iv) showcase how the proposed tuning-free, low memory robust estimator can combat TSAs. Numerical tests with real GPS data demonstrate that accurate time can be provided to the user under various attack conditions.