This paper aims to better analyze the crack resistance of lignin fiber reinforced SMA-13 (LFSMA-13) asphalt mixtures, with and without polymer anti-rut agent (ARA), under different aging degrees. IDEAL-CT test and Fourier transform infrared (FTIR) spectroscopy were utilized to analyze the relationships between the crack resistance of LFSMA-13, with and without ARA, and the parameters of the FTIR spectrum of the asphalt extracted from the test samples. A convenient testing method to predict the anti-crack ability of the mixtures in a road was also derived in this study. The test samples were prepared using the specifications listed by AASHTO. The fracture formation work (Winitial) and cracking index (CTIndex) in the IDEAL-CT test were adopted to reflect the cracking ability of the asphalt mixtures in both the crack formation stage and the crack propagation stage. The peak areas of the FTIR spectrum were utilized to reveal the chemical properties of the asphalt material inside the SMA-13 asphalt mixtures, with and without ARA under different aging degrees. Grey correlation analysis was adopted to choose the most suitable FTIR spectrum parameters to derive the prediction models of Winitial and CTIndex under different aging degrees. After conducting a series of tests, the results showed that the aging process could well affect the crack resistance of the test samples and the peak areas of the asphalt extracted from the mixtures. The FTIR parameters selected from the grey correlation analysis could be used to well predict the anti-crack ability of the asphalt mixtures.