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 quasars 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)
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