  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 analogue alpha Men A
(Teff = 5569 +/- 62 K, R = 0.960 +/- 0.016 Rsun, M = 0.964 +/- 0.045 Msun) as
well as its late M-dwarf companion (Teff = 3142 +/- 86 K, R = 0.24 +/- 0.02
Rsun, M = 0.22 +/- 0.02 Msun). 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 analogue with a
precise asteroseismic age from space-based photometry, making it a prime target
for next-generation direct imaging missions searching for true Earth analogues.
