Are the p-value method and critical-value method supposed to give the same conclusion?

They are two decision routes for the same hypothesis test. If the significance level, tail direction, and test statistic are consistent, both routes should lead to the same reject or do-not-reject conclusion.

For a right-tailed test at the 5 percent level, a test statistic beyond the right-tail critical value implies a p-value below 0.05. If the p-value is 0.02, you reject the null. If the same problem gives the statistic and critical value instead, the statistic should land inside the rejection region. The biggest CFA mistakes are mixing tail direction, comparing the p-value to the confidence level instead of alpha, or treating a high p-value as strong evidence against the null.