Asteroseismology of bright stars has become increasingly important as a method to determine the fundamental properties (in particular ages) of stars. The Kepler Space Telescope initiated a revolution by detecting oscillations in more than 500 main-sequence and subgiant stars. However, most Kepler stars are faint and therefore have limited constraints from independent methods such as long-baseline interferometry. Here we present the discovery of solar-like oscillations in α Men A, a naked-eye (V = 5.1) G7 dwarf in TESS's southern continuous viewing zone. Using a combination of astrometry, spectroscopy, and asteroseismology, we precisely characterize the solar analog α Men A (T eff = 5569 ± 62 K, R ⋆ = 0.960 ± 0.016 R ⊙, M ⋆ = 0.964 ± 0.045 M ⊙). To characterize the fully convective M dwarf companion, we derive empirical relations to estimate mass, radius, and temperature given the absolute Gaia magnitude and metallicity, yielding M ⋆ = 0.169 ± 0.006 M ⊙, R ⋆ = 0.19 ± 0.01 R ⊙, and T eff = 3054 ± 44 K. Our asteroseismic age of 6.2 ± 1.4 (stat) ± 0.6 (sys) Gyr for the primary places α Men B within a small population of M dwarfs with precisely measured ages. We combined multiple ground-based spectroscopy surveys to reveal an activity cycle of P = 13.1 ± 1.1 yr for α Men A, a period similar to that observed in the Sun. We used different gyrochronology models with the asteroseismic age to estimate a rotation period of ~30 days for the primary. Alpha Men A is now the closest (d = 10 pc) solar analog with a precise asteroseismic age from space-based photometry, making it a prime target for next-generation direct-imaging missions searching for true Earth analogs.