No. Let R = k[x, y_i, z_i]/(xy_i, y_iz_i) where i = 1, 2, 3, ... Let S be the multiplicative system regenerated by z_1, z_2, z_3, ... Then x becomes a nonzero divisor in the ring R' = S^{-1}R. Hence the module M' = R'/xR' is pseudo-coherent, i.e., it is finitely n-presented for every n.

But there does not exist a finitely 3-presented module M over R such that M' is the localization of M. First, note that there does exists a finitely presented module, namely R/xR, whose localization is M'. Next, let R_n be the ring where we invert z_1, z_2, ..., z_n in R. Denote M_n = M ⊗ R_n. Because R' = colim R_n for large enough n we have M_n ≅ R_n/xR_n for example by Lemma Tag 05N7 of the Stacks project. Since R ---> R_n is flat, this would imply that R_n/xR_n is finitely 3-presented. But it isn't because the kernel of R_n --- x ---> R_n is not finitely generated.

Remark 1: It is often possible to construct counter examples to (false) general commutative algebra statements by the procedure above.

Remark 2: A true statement is that if one has an (-n)-pseudo coherent module M' over R' and R' is the filtered colimit of rings R_i, then there exists an i and a finite complex of finite free modules E over R_i whose derived base change E' = E ⊗^L_{R_i} R' has cohomology H^0(E') = M' and vanishing H^p(E') for -n < p < 0 and p > 0.

Remark 3: In the localization situation above one can get the complex E of the previous remark to live over the ring R by getting rid of denominators in the obvious way.

No. Let $R = k[x, y_i, z_i]/(xy_i, y_iz_i)$ where $i = 1, 2, 3, \ldots$ Let $S$ be the multiplicative system regenerated by $z_1, z_2, z_3, \ldots$ Then $x$ becomes a nonzero divisor in the ring $R' = S^{-1}R$. Hence the module $M' = R'/xR'$ is pseudo-coherent, i.e., it is finitely $n$-presented for every n.

But there does not exist a finitely 3-presented module $M$ over $R$ such that $M'$ is the localization of $M$. First, note that there does exists a finitely presented module, namely $R/xR$, whose localization is $M'$. Next, let $R_n$ be the ring where we invert $z_1, z_2, \ldots, z_n$ in $R$. Denote $M_n = M \otimes R_n$. Because $R' = \text{colim} R_n$ for large enough $n$ we have $M_n \cong R_n/xR_n$ for example by Lemma Tag 05N7 of the Stacks project. Since $R \to R_n$ is flat, this would imply that $R_n/xR_n$ is finitely $3$-presented. But it isn't, because the kernel of $R_n \stackrel{x}{\to} R_n$ is not finitely generated.

Remark 1: It is often possible to construct counter examples to (false) general commutative algebra statements by the procedure above.

Remark 2: A true statement is that if one has an $(-n)$-pseudo coherent module $M'$ over $R'$ and $R'$ is the filtered colimit of rings $R_i$, then there exists an $i$ and a finite complex of finite free modules $E$ over $R_i$ whose derived base change $E' = E \otimes^L_{R_i} R'$ has cohomology $H^0(E') = M'$ and vanishing $H^p(E')$ for $-n < p < 0$ and $p > 0$.

Remark 3: In the localization situation above one can get the complex $E$ of the previous remark to live over the ring $R$ by getting rid of denominators in the obvious way.