${\mathbb Q}_p = {\mathbb Z} [1/p]$, right? You changed the notation...
Hi, the endomorphisms $E$ have to map the numbers $mp^n$ to some other numbers $mp^n$. Because it's an additive group and we consider homomorphisms, it's enough to figure out what the generators $p^n$ are mapped to for various numbers of $n$. The multiples by $m$ are obtained by the homomorphism conditions.
Moreover, also because it is a homomorphism, $E(p\cdot p^n) = E(p^n+\cdots p^n)$ must be equal to $E(p^n)+E(p^n)+\dots + E(p^n) = p\cdot E(p^n)$ This proves that the information about $E(p^{n})$ and $E(p^{n+1})$ is not independent at all; the latter must be $p$ times the former. We may go in both directions; the former is $1/p$ times the latter.
It follows that the endomorphism is totally determined by $E(p^0)=E(1)$ which has to be an element of ${\mathbb Q}_p$. Note that ${\mathbb Q}_p$ was an additive group so $E(1)$ doesn't have to be $1$. It follows that the endomorphisms as a set are ${\mathbb Q}_p$ and the the endomorphisms, like always, are a semigroup, which is the multiplicative semigroup of these elements, ${\mathbb Q}_p^\times$. The superscript $\times$ means the multiplication. So ${\rm End} {\mathbb Q}_p \equiv {\mathbb Q}_p$.
The automorphisms are those endomorphisms that are one-to-one and therefore invertible. What can $E(1)$ be so that $E(mp^n)$ covers the whole ${\mathbb Q}_p$? If $E(1) = mp^n $ then $ p = (E(1)/m)^{1/n} $ For which $m,n$ is the expression above in ${\mathbb Q}_p$ for any $E(1)$ in ${\mathbb Q}_p$? The $n$th root is clearly too bad and sends us away from ${\mathbb Q}_p$. It can only work for $n=\pm 1$ if $m$ is allowed to be something else than a power of $p$. At any rate, I think that $E(1)$ has to be of the $\pm p^n$ form. The exponent $n$ is an integer so I think that ${\rm Aut} {\mathbb Q}_p \equiv {\mathbb Z} \times {\mathbb Z}_2 $ It's the direct product of the integers (with addition, the group shifting things by a power of $p$) and the two-element group (responsible for the overall sign). (Thanks for the fix, no semidirect here.)