= Solution
Use the standard Euclidean convention $e^{-S_E}$ and restore the curvature normalization explicitly. Let $\varphi$ be the conventional dimensionless <dilaton>, whose worldsheet term is
$$
S_{\varphi}=\frac1{4\pi}\int_\Sigma d^2\sigma\sqrt h\,\varphi R^{(2)}.
$$
For constant $\varphi_0$, the <Gauss-Bonnet theorem> gives $S_{\varphi}=\varphi_0\chi(\Sigma)$. A closed surface of genus $g$ has <Euler characteristic> $\chi=2-2g$, so its contribution is weighted by
$$
e^{-S_{\varphi}}=e^{(2g-2)\varphi_0}=g_s^{2g-2}.
$$
Thus
$$
\boxed{g_s=e^{\varphi_0}.}
$$
This is the <dilaton normalization and Euler-characteristic weighting> of <string perturbation theory>. If the unit curvature coefficient in the printed expression is retained literally in this Euclidean convention, then $\varphi=4\pi\Phi$ and $g_s=e^{4\pi\Phi_0}$. If $\Phi$ denotes the conventional <dilaton>, the usual omitted $1/(4\pi)$ is understood and $g_s=e^{\Phi_0}$. One must specify that normalization before exponentiating the field. Lorentzian overall signs are fixed consistently by Wick rotation.
For the <sigma-model beta function>, use
$$
S_E=\frac1{4\pi\alpha'}\int d^2\sigma\,g_{ab}(X)\partial_\mu X^a\partial_\mu X^b
$$
with vanishing two-form and <dilaton>. Perform the <background field expansion of a string sigma model> covariantly by setting $X=\exp_{\bar X}\eta$. In <Riemann normal coordinates>, the quadratic fluctuation action is
$$
S^{(2)}=\frac1{4\pi\alpha'}\int d^2\sigma
\left[g_{ab}D_\mu\eta^aD_\mu\eta^b
-R_{acbd}\eta^a\eta^b\partial_\mu\bar X^c\partial_\mu\bar X^d\right].
$$
The covariant derivative contains the pulled-back target connection. This expansion accounts for the connection vertices as well as the normal-coordinate metric expansion; keeping only a single metric tadpole would not give a covariant answer.
Integrating the Gaussian fluctuations gives $\frac12\operatorname{Tr}\log\Delta$, where
$$
\Delta^a{}_b=-D^2\delta^a{}_b-R^a{}_{cbd}\partial_\mu\bar X^c\partial_\mu\bar X^d.
$$
Contracting the curvature vertex with the short-distance propagator traces $R^a{}_{cad}$ to the <Ricci tensor> $R_{cd}$. In dimensional regularization at $d=2-\epsilon$, the logarithmic integral is $\int d^dp/(2\pi)^d(p^2+\mu^2)^{-1}=1/(2\pi\epsilon)+O(1)$. With the stated curvature convention the divergent effective action is
$$
\Gamma_{\rm div}^{(1)}=-\frac1{4\pi\epsilon}\int R_{ab}(\bar X)\partial_\mu\bar X^a\partial_\mu\bar X^b.
$$
It is cancelled by the metric counterterm $\delta g_{ab}=\alpha'R_{ab}/\epsilon$. Equivalently, write $g^{\rm bare}=\mu^{-\epsilon}(g+\alpha'R/\epsilon+\cdots)$. Differentiating the bare coupling at fixed value gives the <one-loop metric beta function of a string sigma model>
$$
\boxed{\beta^g_{ab}=\mu\frac{dg_{ab}}{d\mu}=\alpha'R_{ab}+O(\alpha'^2).}
$$
This loop is a loop of two-dimensional fluctuation fields: its expansion is in $\alpha\prime$, not a change of worldsheet genus or an extra power of $g_s$. The sign here defines the renormalization scale to increase toward the ultraviolet. For a round target sphere, positive Ricci curvature makes its squared radius increase with that scale, consistent with the usual asymptotic freedom of the inverse-radius sigma-model coupling. With the printed kinetic coefficient $1/2$ taken literally, $2\pi\alpha'=1$ and the leading coefficient is $R_{ab}/(2\pi)$. Requiring the metric beta function to vanish gives $R_{ab}=0$ at this order. Full bosonic-string Weyl consistency also requires the other anomaly coefficients, including the critical-dimension condition when the <dilaton> vanishes.
For <T-duality>, choose coordinates $(y,x^i)$ adapted to the isometry and use $\alpha'=1$ units for the local <Buscher rules>. Assume $g_{yy}\ne0$, with a spacelike isometry for the usual unitary circle duality. In light-cone worldsheet coordinates take the convention that $E=g+B$ multiplies $\partial_+X^a\partial_-X^b$. Gauge translations of $y$, replace $\partial_\pm y$ by $A_\pm$, gauge fix $y=0$, and impose flatness with a multiplier $\widetilde y$. The relevant first-order Lagrangian is
$$
\begin{aligned}
\mathcal L={}&g_{yy}A_+A_-+g_{yi}(A_+\partial_-x^i+\partial_+x^iA_-)
+g_{ij}\partial_+x^i\partial_-x^j\\
&+\widetilde y(\partial_+A_--\partial_-A_+).
\end{aligned}
$$
Integrating out $\widetilde y$ gives a locally pure-gauge connection and hence the original model. Instead integrate the last term by parts and solve the algebraic equations for $A_\pm$:
$$
A_+=\frac{\partial_+\widetilde y-g_{yi}\partial_+x^i}{g_{yy}},\qquad
A_-=-\frac{\partial_-\widetilde y+g_{yi}\partial_-x^i}{g_{yy}}.
$$
Substitution gives
$$
\begin{aligned}
\mathcal L'={}&\frac1{g_{yy}}\partial_+\widetilde y\partial_-\widetilde y
+\left(g_{ij}-\frac{g_{yi}g_{yj}}{g_{yy}}\right)\partial_+x^i\partial_-x^j\\
&+\frac{g_{yi}}{g_{yy}}\left(\partial_+\widetilde y\partial_-x^i-\partial_+x^i\partial_-\widetilde y\right).
\end{aligned}
$$
Reading its symmetric and antisymmetric parts yields the <Buscher dual of a metric with vanishing two-form>:
$$
\boxed{g'_{yy}=\frac1{g_{yy}},\qquad g'_{yi}=0,\qquad
g'_{ij}=g_{ij}-\frac{g_{yi}g_{yj}}{g_{yy}},}
$$
$$
\boxed{B'_{yi}=\frac{g_{yi}}{g_{yy}},\qquad B'_{iy}=-B'_{yi},\qquad B'_{ij}=0.}
$$
The spectator metric is the <Schur complement> of $g_{yy}$. Reversing the orientation of $\widetilde y$ reverses the displayed mixed two-form signs; our multiplier and $E=g+B$ conventions fix them.
The metric rules arise already from the classical algebraic elimination. For quantum equivalence the regulated Gaussian measure also produces the <Buscher dilaton shift>
$$
\varphi'=\varphi-\frac12\log g_{yy}.
$$
For a compact isometry the multiplier periodicity and flat-connection sectors implement the exchange of momentum and winding. On a circle, restoring dimensions gives \b[$R'=\alpha'/R$]. Thus the calculation derives the local dual background, while the measure and global sectors explain how it becomes a string-theory duality.
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