Let $F:I\to \mathcal{C}$ be a diagram, and consider the colimit $$\varinjlim_{i\in I}\hom_{\mathcal{C}}(F(i),A),$$ where $A$ is a fixed object of $\mathcal{C}$.
Is it considered "legal" to write $\hom_{\mathcal{C}}(F(i),A)=\hom_{\mathcal{C}^{\text{op}}}(A,F(i))$, and use that to rewrite the above colimit as the limit $$\varprojlim_{k\in I^{\text{op}}}\hom_{\mathcal{C}^{\text{op}}}(A^{\text{op}},F^{\text{op}}(k))\;\;?$$
I am just trying to get hold of what is allowed and how I can use the duality principle in this context. Thanks!
It looks like you're confused about variance here. First, the colimit $\varinjlim_{i\in I}\hom_{\mathcal{C}}(F(i),A)$ doesn't make sense because $i\mapsto \hom_{\mathcal{C}}(F(i),A)$ is a contravariant functor from $I$ to $\mathtt{Set}$, not a covariant functor (remember that $\hom$ is contravariant in its first argument and covariant in its second argument). So if you want to be computing a colimit in $\mathtt{Set}$, the index category would be $I^\text{op}$, not $I$.
Now let's look at what happens when we take opposites. As you have said, $\hom_{\mathcal{C}^{\text{op}}}(A^{\text{op}},F^{\text{op}}(k))$ is the same set as $\hom_{\mathcal{C}}(F(k),A)$. But this means that the induced maps between these sets will still go in the same direction. So the canonical functor from $I$ to $\mathtt{Set}$ sending $k$ to $\hom_{\mathcal{C}^{\text{op}}}(A^{\text{op}},F^{\text{op}}(k))$ is still contravariant (you can also see this directly: $F^{\text{op}}$ is contravariant when considered as a functor from $I$ to $\mathcal{C}^\text{op}$, and $\hom$ is covariant in its second argument). When you consider it as a covariant functor $I^\text{op}\to\mathtt{Set}$, this is literally the same functor as the functor $i\mapsto \hom_{\mathcal{C}}(F(i),A)$ we were considering before. Since these are the same functor, we have $$\varinjlim_{i\in I^\text{op}}\hom_{\mathcal{C}}(F(i),A)\cong \varinjlim_{k\in I^{\text{op}}}\hom_{\mathcal{C}^{\text{op}}}(A^{\text{op}},F^{\text{op}}(k)).$$
Notice in particular that a colimit never turned into a limit. The reason is that the difference between a colimit and a limit is the direction of the maps in the category where your diagram lives, which in this case is $\mathtt{Set}$. You didn't change the direction of any maps in $\mathtt{Set}$, you just changed the directions of the maps in the category $\mathcal{C}$ of which you were taking hom-sets.