The copolymer ethylene–octene (POE) has good aging resistance and is an inexpensive asphalt additive compared to the styrene–butadiene–styrene copolymer (SBS). However, POE is easy to segregate in asphalt during storage at high temperatures. Grafting glycidyl methacrylate (GMA) onto the molecular backbone of POE (i.e., POE-g-GMA) may solve this problem, for the epoxy groups in GMA can react with the active groups in asphalt. Asphalt modified with linear and crosslinked POE-g-GMA were prepared, and the hot storage stability, physical properties and thermal oxidation aging properties were discussed in detail. The results show that linear and low-degree crosslinked POE-g-GMA-modified asphalts are storage-stable at high temperatures via measurements of the difference in softening points and small-angle X-ray scattering (SAXS) characterizations from macro and micro perspectives. The difference in softening points (ΔSP) between the upper and lower ends is no more than 3.5 °C for modified asphalts after 48 h of being in an oven at 163 °C. More importantly, the crosslinking modification of POE-g-GMA can further increase the softening point and reduce the penetration as well as rheological properties via conventional physical property, dynamic shear rheometer (DSR) and multiple-stress creep recovery (MSCR) tests. Furthermore, asphalt modified with crosslinked POE-g-GMA reveals better aging resistance via measurements of the performance retention rate and electron paramagnetic resonance (EPR) characterizations after a rolling thin film oven test (RTFOT). This work may provide further guidelines for the application of polymers in asphalt.