Here in the definition 5.3.11 on the page 249 they write "The $p$-value associated with a test is the smallest significance level $α$ for which the null hypothesis is rejected."
My question is what is the order here: $\bar{X}>\bar{x}$ or $\bar{X} <\bar{x}$ and why? I.e. when it is the case that $\alpha$ is the smallest significabnce level, if
$H_0:\mu=\mu_0$ versus $H_1:\mu>\mu_0$
OR on the other hand
$H_0:\mu=\mu_0$ versus $H_1:\mu<\mu_0$ ?
A p-value less than or equal to $\alpha$ means you reject the null hypothesis and have evidence that the mean reflects the alternate hypothesis. The alternate hypothesis can be either $H_a:\mu>\mu_0, \text{ or } H_a : \mu < \mu_0\text{ or simply } H_a : \mu\neq \mu_0$ depending on the context of the test.
For example, you have not tested the mean of a population for several years so you want to assess whether it has changed. Because you don't know whether it is larger or smaller, it is appropriate to use $H_a : \mu\neq \mu_0$. In another situation, you may suspect the mean has decreased (test scores seem to be lower than normal) in which case you would test using $H_a : \mu < \mu_0$.
Edit: One can never prove an $H_a$ with a single test but only provide evidence to support it. One can never discount the possibility of a type I error. This is the random chance of obtaining a p-value less than $\alpha$. There also exists the possibility of another variable affecting the result. But with more testing and control of other variables, one can be more confident in the result.
For $H_a:\mu>\mu_0$, this is a result in the right tail to the right of a Z score corresponding to $\alpha$ of a normal distribution $\text{ and } H_a : \mu < \mu_0$, in the left tail $\text{ and } H_a : \mu\neq \mu_0$ which can be in either tail.