Consider a test for mu. if the p-value is such that you can reject h0 at 5% level of significance, can you always reject h0 at 3% level of significance?

The appropriate response would be "no". Assume the p-esteem is 4%. That is sufficient to dismiss the invalid speculation at the 5% noteworthiness level, however insufficient at 3%. 3% is a higher (lower numerically) hugeness limit. It implies the chances are much lower that the watched result could have happened by shot.