I want to prove the following tautology using resolution:
$$\exists x \forall y P(y,x) \to \forall x \exists y P(x,y)$$
The negation of the sentence in Skolem normal form is
$$\forall y \forall v(P(y,c) \land \neg P(f(y),v))$$
for some function symbol $f$ and constant symbol $c$ and variables $x,y,u,v$
but this seems not unifiable since we can't substitute $y$ for $f(y)$, and thus we can't say the negation of the original sentence is unsatisfiable. Am I missing something here?
Yes, your Skolem form is wrong. The negation of your sentence is $$(\exists x\forall y\,P(y,x))\land (\exists x\forall y\,\lnot P(x,y)).$$ Note that the second $\exists x$ is not within the scope of the first $\forall y$, so in Skolem form we get $$\forall y\forall z\,(P(y,c)\land\lnot P(d,z))$$ where $c$ and $d$ are constants.