We present spatially and spectrally-resolved observations of CH$_3$OH emission from comet C/2012 K1 (PanSTARRS) using The Atacama Large Millimeter/submillimeter Array (ALMA) on 2014 June 28-29. Two-dimensional maps of the line-of-sight average rotational temperature ($T_{rot}$) were derived, covering spatial scales $0.3-1.8$ (corresponding to sky-projected distances $rhosim500$-2500 km). The CH$_3$OH column density distributions are consistent with isotropic, uniform outflow from the nucleus, with no evidence for extended sources of CH$_3$OH in the coma. The $T_{rot}(rho)$ radial profiles show a significant drop within a few thousand kilometers of the nucleus, falling from about 60 K to 20 K between $rho=0$ and 2500 km on June 28, whereas on June 29, $T_{rot}$ fell from about 120 K to 40 K between $rho=$ 0 km and 1000 km. The observed $T_{rot}$ behavior is interpreted primarily as a result of variations in the coma kinetic temperature due to adiabatic cooling of the outflowing gas, as well as radiative cooling of the CH$_3$OH rotational levels. Our excitation model shows that radiative cooling is more important for the $J=7-6$ transitions (at 338 GHz) than for the $K=3-2$ transitions (at 252 GHz), resulting in a strongly sub-thermal distribution of levels in the $J=7-6$ band at $rhogtrsim1000$ km. For both bands, the observed temperature drop with distance is less steep than predicted by standard coma theoretical models, which suggests the presence of a significant source of heating in addition to the photolytic heat sources usually considered.
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