According to this question (or rather answer), a monomial ideal of a polynomial ring is primary iff every variable appearing in it has some power in $I$.
So for example the ideals $(y),(x^3,y^4)$ of $k[x,y]$ are ($P_i$-)primary for some prime ideals $P_i$.
That is, $Ass(k[x,y]/(y))=\{P_1\}$ and $Ass(k[x,y]/(x^3,y^4))=\{P_2\}$.
I looks clear for me that $(y)=Ann_{k[x,y]/(y)}(1)$. So $P_1=(y)$.
But how to find $P_2$? Is it $(x,y)$? If so, how to find an element of the quotient whose annihilator equals $(x,y)$? Is there a general procedure?
$P_2$ must be the radical of $(x^3, y^4)$, so $P_2 = (x,y)$. The annihilator of $x^2 y^3$ in the quotient $k[x,y]/(x^3,y^4)$ is $(x,y)$. You should be able to generalize this.