The dielectric resonances of spherically symmetric micro/nano cavity in zero-index materials have been systematically studied. However, the resonance properties of other shaped dielectric cavities in zero-index materials remain unclear. Here, we theoretically investigate the electromagnetic resonances of the dielectric cavity with cylindrical symmetry in the epsilon-near-zero materials. This kind of cavity supports a set of resonances with strong light confinement, including dipole, quadrupole and higher-order modes with multiple nodes. Furthermore, there is a redshift of the resonance wavelength with an increment of its size, obeying a law as the function of diameter and height. Also, we find that the redshift will be slower for higher-order modes. Through the infinite refractive index contrast and extra degree of freedom, they should have potential application in the enhancement of light-matter interaction and multiple-functional light manipulation in the integrated optical systems.