Let $T$ be a topos, and $F \in T$, $T/F$ a localization of $T$. So we have a natural morphism $i: T/F \longrightarrow T$.
My questions are:
1.What are the definitions of $i_{\ast}$ and $i^{\ast}$ without using site ?
2.Dose there exist a site $S$ such that $Sh(S) \cong T$ and there is an $U \in S$ which $U^{\sim} \cong F$ ?
$i^*$ is the functor which send an object $X \in T$ to $X \times F$ with the natural projection as map into $F$.
$i_*$ is a little harder to describe, if $p: Y \rightarrow F$ is an object of $T/F$, then $i_*(Y)$ is the sub-object of $[F,Y]$ (the internal hom object) which corresponds to map $f$ from $F$ to $Y$ such that $p \circ f =Id\_F$ this can be express as an equaliser or with internal language as you prefers)
You also have a $i\_!$ functor (left adjoint to $i^*$ ) which is just the forget functor who send $p:Y \rightarrow F$ to $Y$.
For your second question :
You can chose any generating family $B$ of $T$ (for example the image by the yoneda embeddings of a site of definition of $T$) add $F$ to this familly, the familly $B'$ you obtain seen as a full subcategory of $T$ and endowed with the canonical topology of $T$ is (by the Grothendieck comparison lemma) a site of definition of $T$, and you simply choose $U$ to be $F$.
