RX J0046.5$-$7308 is a shell-type supernova remnant (SNR) in the Small Magellanic Cloud (SMC). We carried out new $^{12}$CO($J$ = 1-0, 3-2) observations toward the SNR using Mopra and the Atacama Submillimeter Telescope Experiment. We found eight molecular clouds (A-H) along the X-ray shell of the SNR. The typical cloud size and mass are $sim$10-15 pc and $sim$1000-3000 $M_{odot}$, respectively. The X-ray shell is slightly deformed and has the brightest peak in the southwestern shell where two molecular clouds A and B are located. The four molecular clouds A, B, F, and G have high intensity ratios of $^{12}$CO($J$ = 3-2) / $^{12}$CO($J$ = 1-0) $> 1.2$, which are not attributable to any identified internal infrared sources or high-mass stars. The HI cavity and its expanding motion are found toward the SNR, which are likely created by strong stellar winds from a massive progenitor. We suggest that the molecular clouds A-D, F, and G and HI clouds within the wind-blown cavity at $V_mathrm{LSR} = 117.1$-122.5 km s$^{-1}$ are to be associated with the SNR. The X-ray spectroscopy reveals the dynamical age of $26000^{+1000}_{-2000}$ yr and the progenitor mass of $gtrsim 30$ $M_{odot}$, which is also consistent with the proposed scenario. We determine physical conditions of the giant molecular cloud LIRS 36A using the large velocity gradient analysis with archival datasets of the Atacama Large Millimeter/submillimeter Array; the kinematic temperature is $72^{+50}_{-37}$ K and the number density of molecular hydrogen is $1500^{+600}_{-300}$ cm$^{-3}$. The next generation of $gamma$-ray observations will allow us to study the pion-decay $gamma$-rays from the molecular clouds in the SMC SNR.
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