The $^{18}$Ne($alpha$,$p$)$^{21}$Na reaction provides a pathway for breakout from the hot CNO cycles to the $rp$-process in type I x-ray bursts. To better determine this astrophysical reaction rate, the resonance parameters of the compound nucleus $^{22}$Mg have been investigated by measuring the resonant elastic scattering of $^{21}$Na+$p$. An 89 MeV $^{21}$Na radioactive ion beam was produced at the CNS Radioactive Ion Beam Separator and bombarded an 8.8 mg/cm$^2$ thick polyethylene target. The recoiled protons were measured at scattering angles of $theta_{c.m.}$$approx 175 {^circ}$ and 152${^circ}$ by three $Delta E$-$E$ silicon telescopes. The excitation function was obtained with a thick-target method over energies $E_x$($^{22}$Mg)=5.5--9.2 MeV. The resonance parameters have been determined through an $R$-matrix analysis. For the first time, the $J^{pi}$ values for ten states above the alpha threshold in $^{22}$Mg have been experimentally determined in a single consistent measurement. We have made three new $J^{pi}$ assignments and confirmed seven of the ten tentative assignments in the previous work. The $^{18}$Ne($alpha$,$p$)$^{21}$Na reaction rate has been recalculated, and the astrophysical impact of our new rate has been investigated through one-zone postprocessing x-ray burst calculations. We find that the $^{18}$Ne($alpha$,$p$)$^{21}$Na rate significantly affects the peak nuclear energy generation rate and the onset temperature of this breakout reaction in these phenomena.
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