Homotopy pullback of $\mathbb{A}^1$-projections in the Nisnevich localization Let $L_{Nis}(sPre(Sm_S))$ be the Nisnevich localization of the category of simplicial presheaves, 
how to see that whether $\mathbb{A}^1$-projections  $\mathbb{A}^1\times_S X\to X$ are closed under homotopy pullback in $L_{Nis}(sPre(Sm_S))$? How to compute the homotopy pullback of maps $\mathbb{A}^1\times_S X\to X$ along itself in $L_{Nis}(sPre(Sm_S))$? 
 A: First, in any model category, weak equivalences are closed under homotopy pullback. That's the whole point of the "homotopy" in "homotopy pullback." 
In your question, you are missing a step. Generally, $Sm_S$ is the category of smooth and finite type schemes over a scheme S of finite dimension. This category doesn't have good categorical behavior (it's not bicomplete, for example), so Voevodsky embedded it into simplicial presheaves $SPre(Sm_S)$, i.e. functors into simplicial sets. This has a model structure and you can do the localization $L_{Nis} SPre(Sm_S)$ that you wanted. To get the statement about $\mathbb{A}^1$-weak equivalences, you do a further localization to make $\mathbb{A}^1$ contractible, and it's this model structure that tells you homotopy pullbacks preserve weak equivalences. See Morel-Voevodsky.
The resulting model category is proper, meaning that if you look at a diagram $A \to B \gets C$ where one of those maps is a fibration, then you can compute homotopy pullback as just the regular old pullback, which is easy in presheaf categories. For a proof, see Jardine. If neither of those maps is a fibration, you can do fibrant replacement to replace one by a fibration.
