  Asteroseismology of bright stars has become increasingly important as a
method to determine 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 $\alpha$ 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 alpha Men A (Teff
= 5569 +/- 62 K, R = 0.960 +/- 0.016 Rsun, M = 0.964 +/- 0.045 Msun). 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, R = 0.19 +/- 0.01 and
Teff = 3054 +/- 44 K. Our asteroseismic age of 6.2 +/- 1.4 (stat) +/- 0.6 (sys)
Gyr for the primary places $\alpha$ 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 13.1 +/- 1.1 years, 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=10pc) 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.
