We present results from the Hitomi X-ray observation of a young composite-type supernova remnant (SNR) G21.5-0.9, whose emission is dominated by the pulsar wind nebula (PWN) contribution. The X-ray spectra in the 0.8-80 keV range obtained with the Soft X-ray Spectrometer (SXS), Soft X-ray Imager, and Hard X-ray Imager (HXI) show a significant break in the continuum as previously found with the NuSTAR observation. After taking into account all known emissions from the SNR other than the PWN itself, we find that the Hitomi spectra can be fitted with a broken power law with photon indices of Gamma(1) = 1.74 +/- 0.02 and Gamma(2) = 2.14 +/- 0.01 below and above the break at 7.1 +/- 0.3 keV, which is significantly lower than the NuSTAR result (similar to 9.0 keV). The spectral break cannot be reproduced by time-dependent particle injection one-zone spectral energy distribution models, which strongly indicates that a more complex emission model is needed, as suggested by recent theoretical models. We also search for narrow emission or absorption lines with the SXS, and perform a timing analysis of PSR J1833-1034 with the HXI and the Soft Gamma-ray Detector. No significant pulsation is found from the pulsar. However, unexpectedly, narrow absorption line features are detected in the SXS data at 4.2345 keV and 9.296 keV with a significance of 3.65 sigma. While the origin of these features is not understood, their mere detection opens up a new field of research and was only possible with the high resolution, sensitivity, and ability to measure extended sources provided by an X-ray microcalorimeter.