For the catenoid $C$ parametrized by $r(u,v)= (u, \cosh u \cos v, \cosh u \sin v)$, we have $r_u(u,v)= (1, \sinh u \cos v, \sinh u \sin v),$
$r_v(u,v)= (0, -\cosh u \sin v, \cosh u \cos v).$ This implies $r_u\times r_v=(\sinh u\cosh u,-\cosh u\cos v,-\cosh u \sin v).$
This gives $|r_u\times r_v|=\cosh u\sqrt{(\sinh u)^2+1}=(\cosh u)^2$ and $N=\frac{r_u\times r_v}{|r_u\times r_v|}=\frac{1}{\cosh u}(\sinh u,-\cos v,-\sin v).$ I think this is what got for the unit normal.
Now, fixed a point $p\in C$ and assume $p=r(u_0,v_0)$. Then the coordinate curves at $p$ are given by $\alpha:(-\epsilon,\epsilon)\rightarrow C$ and $\beta:(-\epsilon,\epsilon)\rightarrow C$ such that $\alpha(t)=r(u_0+t,v_0)$ and $\alpha(t)=r(u_0,v_0+t)$ are coordinate curves at $p$, because $\alpha(0)=\beta(0)=r(u_0,v_0)=p$. Then \alpha'(0)=\frac{d}{dt}r(u_0,v_0+t)\Big|_{t=0}=r_u(u_0,v_0)\in T_pC, \beta'(0)=\frac{d}{dt}r(u_0+t,v_0)\Big|_{t=0}=r_v(u_0,v_0)\in T_pC.
Differentiate $N(\alpha(t))$ with respect to $t$ and using $(2)$, we have $\frac{d}{dt}N(\alpha(t))\Big|_{t=0}=\frac{d}{dt}\left(\frac{1}{\cosh(u_0+t)}(\sinh(u_0+t),-\cos v_0,-\sin v_0)\right)\Big|_{t=0}$ $=\left(-\frac{\sinh(u_0+t)}{\cosh^2(u_0+t)}(\sinh(u_0+t),-\cos v_0,-\sin v_0) \right)\Big|_{t=0}$ $+\left(\frac{1}{\cosh(u_0+t)}(\cosh(u_0+t),0,0)\right)\Big|_{t=0}$ $=\frac{1}{\cosh^2(u_0)}(1,\sinh u_0\cos v_0,\sinh u_0\sin v_0).$ Using chain rule, the left hand side is dN_{\alpha(0)}(\alpha'(0))=dN_{p}(\alpha'(0)). On the other hand, the right hand side is equal to \displaystyle\frac{1}{\cosh^2(u_0)}\alpha'(0) by $(1)$ and $(3)$. Therefore, we have shown that dN_{p}(\alpha'(0))=\frac{1}{\cosh^2(u_0)}\alpha'(0), i.e. the coordinate curve $\alpha$ is principal direction with principal curvature $\cosh^2(u_0)$.
Similarly you can show that the coordinate curve $\beta$ is principal direction. I will let you do it.