What kind of methods there are to find integer solutions of $x^3+y^3-xy^2=1$? I tried some inequalities and congruences without success. I also found on Wikipedia that this might be a Thue equation but I have no idea what is a bivariate form.
You are correct, this is a Thue equation. "bivariate" just means "having two variables", and "form" refers to the polynomial being homogeneous, which it is since every term has degree 3. Consequently, there are only finitely many solutions, but finding them all can be quite involved. The general procedure requires working over $\mathbf Q(\alpha)$ where $\alpha$ in this case is a root of $x^3-x+1$. You might want to try the Thue solver in Pari/GP, rather than carry this out directly.
The fact that (1,0),(0,1),(1,1) are solutions suggests you wouldn't be able to rule out too much by congruences.