The Crab nebula originated from a core-collapse supernova (SN) explosion observed in 1054 AD. When viewed as a supernova remnant (SNR), it has an anomalously low observed ejecta mass and kinetic energy for an Fe-core-collapse SN. Intensive searches have been made for a massive shell that solves this discrepancy, but none has been detected. An alternative idea is that SN 1054 is an electron-capture (EC) explosion with a lower explosion energy by an order of magnitude than Fe-core-collapse SNe. X-ray imaging searches were performed for the plasma emission from the shell in the Crab outskirts to set a stringent upper limit on the X-ray emitting mass. However, the extreme brightness of the source hampers access to its vicinity. We thus employed spectroscopic technique using the X-ray micro-calorimeter on board the Hitomi satellite. By exploiting its superb energy resolution, we set an upper limit for emission or absorption features from as yet undetected thermal plasma in the 2-12 keV range. We also re-evaluated the existing Chandra and XMM-Newton data. By assembling these results, a new upper limit was obtained for the X-ray plasma mass of less than or similar to 1 M-circle dot for a wide range of assumed shell radius, size, and plasma temperature values both in and out of collisional equilibrium. To compare with the observation, we further performed hydrodynamic simulations of the Crab SNR for two SN models (Fe-core versus EC) under two SN environments (uniform interstellar medium versus progenitor wind). We found that the observed mass limit can be compatible with both SN models if the SN environment has a low density of less than or similar to 0.03 cm(-3) (Fe core) or less than or similar to 0.1 cm(-3) (EC) for the uniform density, or a progenitor wind density somewhat less than that provided by amass loss rate of 10(-5) M-circle dot yr(-1) at 20 km s(-1) for the wind environment.