Should the p-value approach and critical-value approach always agree?

Yes, they should agree if you use the same alternative hypothesis, significance level, and tail direction.

For a right-tailed 5% test:

• Critical-value approach: reject if the statistic is greater than the right-tail critical value.
• P-value approach: reject if the right-tail p-value is less than 5%.

The approaches can seem inconsistent when the tail is mixed up. For example, using a one-tailed p-value for a two-tailed test can make the evidence look stronger than it really is.

So if your answers conflict, check the tail before checking the arithmetic.