Proving adjoint functors preserve limits by lifting the adjunction to cone categories

It is indeed true that $F'\dashv G'$, and not too hard to show once you have the right picture in mind.

The thing to notice is that an object in $D'$ is a cone over $Q$, i.e. an object $y$ of $D$ together with a natural transformation $\beta:\Delta_D(y)\Rightarrow Q$, and an arrow $(y,\beta,Q)\to (y',\beta',Q)$ is just an arrow $g:y\to y'$ in $D$ that commutes with $\beta$ and $\beta'$, in the sense that $\beta_i'\circ g=\beta_i$ for all objects $i$ of $I$. $C'$ can be described similarly.

In particular, an arrow $(x,\alpha,GQ)\to (Gy,G\beta,GQ)$ is just an arrow $h:x\to Gy$ such that $G\beta_i\circ h=\alpha_i$, and an arrow $(Fx,\alpha^{ad},Q)\to (y,\beta,Q)$ is just an arrow $k:Fx\to y$ such that $\beta_i\circ k =\alpha_i^{ad}$; so the bijection induced by the adjunction $F\dashv G$ restricts (by naturality) to a bijection between arrows of one kind and arrows of the other kind, and it is still natural, since $F'$ and $G'$ acts on arrows like $F$ and $G$. Hence $F'\dashv G'$.

This answer is meant to illustrate "another route" that "might be preferable".

First, we need a convenient characterizations of limits. The most flexible one for our purposes is via representability: $\mathsf{Hom}(X,\mathsf{Lim}_{I\in\mathcal{I}}D(I))\cong\mathsf{Lim}_{I\in\mathcal{I}}\mathsf{Hom}(X,D(I))$ natural in $X$. This can be taken as the definition of what a limit is assuming you know what limits are in $\mathbf{Set}$. Alternatively, you can easily prove this fact from other characterizations of limits.

The calculation that right adjoints preserves limits is then simple and direct: $$\begin{align} \mathsf{Hom}(X,G(\mathsf{Lim}_{I\in\mathcal{I}}D(I))) & \cong \mathsf{Hom}(F(X),\mathsf{Lim}_{I\in\mathcal{I}}D(I)) \\ & \cong \mathsf{Lim}_{I\in\mathcal{I}}\mathsf{Hom}(F(X),D(I)) \\ & \cong \mathsf{Lim}_{I\in\mathcal{I}}\mathsf{Hom}(X,G(D(I))) \\ & \cong \mathsf{Hom}(X,\mathsf{Lim}_{I\in\mathcal{I}}G(D(I))) \end{align}$$ The desired result then follows by Yoneda. $\square$

Going one step further in our characterization of limits, we can calculate (using ends and the naturality formula or directly) that $\mathsf{Nat}(\Delta X,D)\cong\mathsf{Hom}(X,\mathsf{Lim}_{I\in\mathcal{I}}D(I))$ which is to say, if this is additionally natural in $D$, that $\Delta \dashv \mathsf{Lim}$. We can take this adjunction as our characterization of having limits of shape $\mathcal{I}$. (Note, before we were only talking about a specific limit of $D$, now we're talking about having the limit of any diagram of shape $\mathcal{I}$.) This characterization produces a basically identical derivation: $$\begin{align} \mathsf{Hom}(X,G(\mathsf{Lim}(D))) & \cong \mathsf{Hom}(F(X),\mathsf{Lim}(D)) \\ & \cong \mathsf{Nat}(\Delta(F(X)),D) \\ & = \mathsf{Nat}(F\circ\Delta(X),D) \\ & \cong \mathsf{Nat}(\Delta(X),G\circ D) \\ & \cong \mathsf{Hom}(X,\mathsf{Lim}(G\circ D)) \end{align}$$ $\square$