  Methanol is an important tracer to probe physical and chemical conditions in
the interstellar medium of galaxies. Methanol is also the most sensitive target
molecule for probing potential space-time variations of the proton-electron
mass ratio, mu, a dimensionless constant of nature. We present an extensive
ALMA study of the strongest submillimeter absorption lines of methanol [...] in
the z=0.89 molecular absorber toward PKS1830-211, the only high-redshift object
in which methanol has been detected. Our goals are to constrain the excitation
of the methanol lines and to investigate the cosmological invariance of mu
based on their relative kinematics. [...] We explore methanol excitation by
running the non local thermal equilibrium radiative transfer code RADEX [...]
The excitation analysis points to a cool (~10-20 K) and dense (~10^{4-5} cm-3)
methanol gas. [...] In addition, we measure an abundance ratio A/E = 1.0 +/-
0.1, an abundance ratio CH3OH/H2 ~ 2 x 10^{-8}, and a 12CH3OH/13CH3OH ratio 62
+/- 3. Our analysis shows that the bulk velocities of the different transitions
are primarily correlated with the observing epoch due to morphological changes
in the background quasar's emission. There is a weaker correlation between bulk
velocities and the lower level energies of the transitions, which could be a
signature of temperature-velocity gradients in the absorbing gas. As a result,
we do not find evidence for variations of mu, and we estimate Dmu/mu = (-1.8
+/- 1.2) x 10^{-7} at 1sigma from our multivariate linear regression. We set a
robust upper limit | \Dmu/mu | < 3.6 x 10^{-7} (3sigma) for the invariance of
mu at a look-back time of half the present age of the Universe. Our analysis
highlights that systematics need to be carefully taken into account in future
radio molecular absorption studies aimed at testing Dmu/mu below the 10^{-7}
horizon. (Abridged)
