$\mathbb{R}^2_+$ and $U$ are subsets of $\mathbb{C}$ with:
$\mathbb{R}^2_+ = \left \{ (x,y) \in \mathbb{R}^2 : y > 0 \right \}$
$U = \left \{ (u,v) \in \mathbb{R}^2 : u^2 +v^2 < 1 \right \}$
Show that through $\varphi (z) = \frac{z-i}{z+1}$ a Diffeomorphism $\mathbb{R}^2_+ \rightarrow U$ is defined, with respect to the respective hyperbolic metric is an isometry.
I calculated the metric tensors:
$g_{ij}(U) = \begin{pmatrix}
\frac{4}{(1-u^2-v^2)^2} &0 \\
0 & \frac{4}{(1-u^2-v^2)^2}
\end{pmatrix}$
and $g_{ij}(\mathbb{R}^2_+) = \begin{pmatrix}
\frac{1}{y^2} &0 \\
0 & \frac{1}{y^2}
\end{pmatrix}$
However, I dont know how to show that its an isometry.
The map that you wrote is not an isometry, I'll give you the correct definition. It's a very long calculation but not so difficult.
Recall that a smooth map $f:(X,g)\rightarrow (Y,h)$ between riemannian manifold is an isometry if it is a diffeomorphism and $f^*h=g$, where $\forall u,v\in T_pX$ one has $f^*h(u,v):=h(d_pf(u),d_pf(v))$ and $d_pf: T_pX\rightarrow T_{f(p)}Y$ is the differential of $f$ at $p$.
Step 1:
Let $\mathbb{H}=\{(u,v)\in\mathbb{R}^2 | v>0 \}$ with metric $g=\frac{du^2+dv^2}{v^2}$ and let $\mathbb{B}=\{(x,y) | \sqrt{x^2+y^2}<1\}$ with metric $h=\frac{4}{1-x^2-y^2}(dx^2+dy^2)$. Define the map $\displaystyle f:\mathbb{H}\rightarrow\mathbb{B}$ to be $f(z):=\frac{z-i}{z+i}$ where I'm identifying $z=x+iy$. The first thing one has to show is that Im$(f)\subset\mathbb{B}$, i.e. that for all $z\in\mathbb{H}$ then $f(z)\in\mathbb{B}$. This is equivalent to prove that if $\Im(z)>0$ then $|z-i|<|z+i|$ and you can check it by yourself.
Step 2:
$f$ is a diffeomorphism (notice that $-i\notin\mathbb{H}$).
Check that the map $\sigma:\mathbb{B}\rightarrow\mathbb{H}$ given by $\sigma(w):=\frac{w+1}{i(w-1)}$ is a well defined smooth map and inverse of $f$.
Step 3:
$f^*h=g$ which implies that $f$ is an isometry between the hyperbolic plane and the Poincarè disk.
To do this just write $h$ and $g$ in terms of complex numbers $z=x+iy$ and $w=u+iv$ and then use the definition of the pullback metric above.
Try to do this by your own it's a very useful exercise!