To investigate the missing compact star of Supernova 1987A, we analyzed the cooling and heating processes of a possible compact star based on the upper limit of observational X-ray luminosity. From the cooling process, we found that a solid quark-cluster star (SQS), having a stiffer equation of state than that of a conventional liquid quark star, has a heat capacity much smaller than a neutron star. The SQS can cool down quickly, naturally explaining the non-detection of a point source in X-ray wavelengths. On the other hand, we considered the heating processes due to magnetospheric activity and possible accretion and obtained some constraints on the parameters of a possible pulsar. Therefore, we concluded that a SQS can explain the observational limit in a confident parameter space. As a possible central compact object, the pulsar parameter constraints can be tested for SN1987A with advanced, future facilities.