We carried out new $^{12}$CO($J$ = 1-0, 3-2) observations of a N63A supernova remnant (SNR) from the LMC using ALMA and ASTE. We find three giant molecular clouds toward the northeast, east, and near the center of the SNR. Using the ALMA data, we spatially resolved clumpy molecular clouds embedded within the optical nebulae in both the shock-ionized and photoionized lobes discovered by previous H$alpha$ and [S II] observations. The total mass of the molecular clouds is $sim$$800$ $M_{odot}$ for the shock-ionized region and $sim$$1700$ $M_{odot}$ for the photoionized region. Spatially resolved X-ray spectroscopy reveals that the absorbing column densities toward the molecular clouds are $sim$$1.5$-$6.0times10^{21}$ cm$^{-2}$, which are $sim$$1.5$-$15$ times less than the averaged interstellar proton column densities for each region. This means that the X-rays are produced not only behind the molecular clouds, but also in front of them. We conclude that the dense molecular clouds have been completely engulfed by the shock waves, but have still survived erosion owing to their high-density and short interacting time. The X-ray spectrum toward the gas clumps is well explained by an absorbed power-law or high-temperature plasma models in addition to the thermal plasma components, implying that the shock-cloud interaction is efficiently working for both the cases through the shock ionization and magnetic field amplification. If the hadronic gamma-ray is dominant in the GeV band, the total energy of cosmic-ray protons is calculated to be $sim$$0.3$-$1.4times10^{49}$ erg with the estimated ISM proton density of $sim$$190pm90$ cm$^{-3}$, containing both the shock-ionized gas and neutral atomic hydrogen.
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