The question states the following:
Let $\mathscr{C}$ and $J$ be categories, and let $\triangle:\mathscr{C}\to\mathscr{C}^J$ be the functor that sends an object $C\in\mathscr{C}$ to the constant diagram valued at $C$. Suppose that $\triangle$ has a right adjoint $R:\mathscr{C}^J\to\mathscr{C}$, then every $J$-shaped diagram $F:J\to\mathscr{C}$ in $\mathscr{C}$ has a limit.
My attempt of the proof claims that $RF\in\mathscr{C}$ (I'm not even sure if this is correct right now) is a limit of arbitrary $J$-shaped diagram $F:J\to\mathscr{C}$, and tries to check the definition of a cone and the universal property. My problem with this attempt is that it doesn't really use the fact that there is the isomorphism $\mathscr{C}^J(\triangle C,F)\cong \mathscr{C}(C,RF)$ for any $C\in\mathscr{C}$. So I had been really skeptical about my proof. I tried to use the idea that the limits is basically a representation $\mathscr{C}(-,\lim\limits_{J}F)\cong \text{Cone}(-,F)$, but I think that needs the premise on the size of categories I have. So what did I miss?
The short answer is that for a diagram $F: J \to \mathcal{C}$ a cone $(q_j: C \to F(j))_{j \in J}$ is the same data as a natural transformation $q: \Delta C \to F$. So by the adjunction $\mathcal{C}^J(\Delta C, F) \cong \mathcal{C}(C, RF)$, cones correspond bijectively to arrows $C \to RF$.
We can be a bit more precise. To specify the limit you will also need to specify the projections. For this we can use the counit $\varepsilon: \Delta R \to Id_{\mathcal{C}^J}$ of the adjunction. That is, given a diagram $F: J \to \mathcal{C}$ we can specify a cone with vertex $RF$ and then for each $j$ we get an arrow $\varepsilon(F)(j): R F \to F(j)$. The fact that this is indeed a cone follows from $\varepsilon(F): \Delta R F \to F$ being a natural transformation. This will be our limit.
There is one thing left to check to finish the argument. Let $(q_j: C \to F(j))_{j \in J}$ be any cone, which we consider as a natural transformation $q: \Delta C \to F$. So under the adjunction $\mathcal{C}^J(\Delta C, F) \cong \mathcal{C}(C, RF)$ this $q$ corresponds to a unique arrow $u: C \to RF$. This is of course going to be the unique arrow that makes $RF$ satisfy the universal property, but this needs some checking (hint: use the way the unit and counit interact in an adjunction).