As I understood, a minimal polynomial for some $A$ over a finite field is an irreducible polynomial with a minimal degree for which $A$ is a "root".
However, I can't understand how to find one. For example, if we have an element of order 13 over $GF(27)$. (I want to find a minimal polynomial for such an $A$)
If $\alpha$ has order 13, then it is a root of $x^{13}-1$. But this polynomial is not irreducible. So you need to find the irreducible factor of $x^{13}-1$ that has $\alpha$ as a root.
For your particular example, if you take $\alpha$ to be the generator of $GF(27)$, then $\alpha$ has order 26. Then $\alpha^{2}$ will have order 13.
When you talk about minimal polynomials you want to know over which field. For example, an element $\beta$ of order 13 is in $GF(27)$, so the minimal polynomial over $GF(27)$ will be $x-\beta$. On the other hand, if you want to find the minimal polynomial of $\beta$ over $GF(3)$, it will be something different.
For more general examples, if you look up cyclotomic cosets you will find a method for factoring a polynomial like this, and if you look up cyclotomic polynomials it should help you find the factor you are looking for.