What would an example of a null and alternative hypothesis look like for a one right tail test with a rejection region at an alpha of 1 percent?

It depends on the statistical test that is being applied. Different tests have different critical values. For the sake of definiteness let's say that you plan to measure the IQs of 30 people using two different paper-and-pencil tests. In advance of your study you decided that the null hypothesis would be that there is no correlation between the two tests and that the alternative hypothesis would be that a positive correlation of the results of test B on those of test A would be accepted at level p=0.01 (or 1%).

You proceed with your measurements and you make your correlation calculation. Meanwhile you have consulted the necessary tables to find that the critical value for the correlation statistic based on 30 points is 0.306.

This means that you must accept the null hypothesis if the sample correlation coefficient is less than or equal to 0.306. You can accept the alternative if the sample statistic is greater than 0.306.