I am following a mixed convention between Hartshorne AG and Tennison's sheaf theory.
Let $\phi:F\to G$ be a morphism of presheaves $F,G$ over topological space $X$. The standard assertion is that $\phi$ is isomorphism iff $\phi_p$ is isomorphism for each $p\in X$ where $\phi_p$ denotes the induced stalk level morphism.
I want to break the assertion into two parts, one for injectivity and one for surjectivity, especially for the converse implication. The forward implication does not care whether $F,G$ are sheaves. One can assume I am working in the category of sheaves of abelian groups over $X$ and $F,G$ are objects in this category. The argument should not be significantly different in other category of sheaves.
For injectivity, I require $F$ being a sheaf and $G$ being a monosheaf(i.e. $s\in G(U)$ if $s|_{U_i}=0$ for $U_i$ covering $U$, then $s=0$). If $\phi_p$ is injection for each $p\in X$, then $\phi$ is injection.
For surjectivity, it seems there is no way to weakening the assumption on $F$ and $G$ being sheaves to deduce surjectivity from stalk map.
Q. Is above observation correct?
So the standard statement is : a morphism of sheaves is an injection/surjection/isomorphism iff it is true stalkwise.
This does not works for presheaves as you observed it. But you can still deduce some facts. So assume you proved the standard statement for sheaves and let's see what is going on for presheaves.
We will use the associated sheaf functor which will be denoted by $aF$.
If $\phi:F\rightarrow G$ is a morphism of presheaves, such that $\phi_p$ is an injection for every $p$, then $aF\rightarrow aG$ is an injection and $aF(U)\rightarrow aG(U)$ is into for every $U$. If $F(U)\rightarrow aF(U)$ is into, from the commutative square $$\require{AMScd} \begin{CD} F(U)@>>> G(U)\\ @VVV@VVV\\ aF(U)@>>> aG(U) \end{CD} $$ you can deduce that $F\rightarrow G$ is injective. But $F\rightarrow aF$ is into iff $F$ is separated (monosheaf with your terminology). So for injectivity, you only need $F$ separated.
Hence $F$ separated, $G$ arbitrary, and $\phi_p$ into at every $p$ $\Rightarrow \phi$ is an injection.
For surjectivity, we must be very careful. Surjectivity for sheaves and for presheaves does not mean the same thing !
In the standard statement, surjectivity is meant for sheaves. Hence this is not true that $\phi_p$ onto at every $p$ implies that for all open $U$, $F(U)\rightarrow G(U)$ is onto.
So a more general statement with presheaves seems very unlikely to hold since it won't hold for sheaves ! Unless you mean surjectivity for sheaves, but then it doesn't mean anything if we are not considering sheaves...