The first detailed study on the performance of a ZnO-based cryogenic scintillating bolometer as a detector to search for rare processes in zinc isotopes was performed. A 7.2 g ZnO low-temperature detector, containing more than 80% of zinc in its mass, exhibits good energy resolution of baseline noise 1.0--2.7 keV FWHM at various working temperatures resulting in a low-energy threshold for the experiment, 2.0--6.0 keV. The light yield for $\beta$/$\gamma$ events was measured as 1.5(3) keV/MeV, while it varies for $\alpha$ particles in the range of 0.2--3.0 keV/MeV. The detector demonstrate an effective identification of the $\beta$/$\gamma$ events from $\alpha$ events using time-properties of only heat signals. %(namely, Rise time parameter). The radiopurity of the ZnO crystal was evaluated using the Inductively Coupled Plasma Mass Spectrometry, an ultra-low-background High Purity Ge $\gamma$-spectrometer, and bolometric measurements. Only limits were set at the level of $\mathcal{O}$(1--100) mBq/kg on activities of \Nuc{K}{40}, \Nuc{Cs}{137} and daughter nuclides from the U/Th natural decay chains. The total internal $\alpha$-activity was calculated to be 22(2) mBq/kg, with a major contribution caused by 6(1) mBq/kg of \Nuc{Th}{232} and 12(2) mBq/kg of \Nuc{U}{234}. Limits on double beta decay (DBD) processes in \Nuc{Zn}{64} and \Nuc{Zn}{70} isotopes were set on the level of $\mathcal{O}(10^{17}$--$10^{18})$ yr for various decay modes profiting from 271 h of acquired background data in the above-ground lab. This study shows a good potential for ZnO-based scintillating bolometers to search for DBD processes of Zn isotopes, especially in \Nuc{Zn}{64}, with the most prominent spectral features at $\sim$10--20 keV, like the two neutrino double electron capture. A 10 kg-scale experiment can reach the experimental sensitivity at the level of $\mathcal{O}(10^{24})$ yr.