There is also an algebraic proof which works over fields of positive characteristic.

See, e.g., Lemma 5.1 of the paper Zariski's conjecture and related problems by Madhav Nori (Annales scientifiques de l'École Normale
Supérieure, Sér. 4, 16 no. 2 (1983), p. 305-344).

The proof of that Lemma is more general than what you need. It goes as follows. Let $X$ be a proper smooth algebraic surface and let $A\subset X$ be an ample curve. That $\pi_1(A)$ surjects onto $\pi_1(X)$ means the following: any finite étale cover $f:Y\to X$ which is split over $A$ is split. So if $Y$ is connected, $\deg(f)=1$.

Let us prove that if $A$ is ample; in fact, we only need $A$ big and nef.
If $f$ has a section on $A$, one can write $f^{-1}(A)=B+R$ where $B\to A$
is an isomorphism, and $R$ is disjoint from $B$. The Hodge index theorem
says that the intersection form restricted to the space generated by $B$ and $R$ has
at most one +-sign. Since $(B+R)^2=(f^*A)^2=\deg(f) A^2>0$, it has exactly one +-sign,
and the determinant
$$\begin{vmatrix} (B+R)^2 & (B+R)\cdot B \cr B\cdot (B+R) & B^2\end{vmatrix}
$$
is nonpositive. By the projection formula, one has
$$(B+R)\cdot B=f^*A\cdot B=A\cdot f_*B=A^2.$$
Since $B$ and $R$ are disjoint, we obtain $B^2=A^2$. Then,
$$ \deg(f)A^2 = (B+R)^2=B^2+R^2, $$
so $$R^2=(\deg(f)-1) A^2$$.
The above determinant is equal to
$ (\deg(f)-1)A^2$. Since $A^2>0$, $\deg(f)\leq 1$.

This proof generalizes to the so-called Ramanujam lemma according to which an effective divisor on a surface which is big and nef is numerically connected (doesn't decompose as the sum of two nonzero effective divisors with 0-intersection), hence connected. In our case, the existence of the section implies that $f^*A$ is not numerically connected; it is however big and nef because this property is stable under finite pull-back.

See also the paper of J-B. Bost, Potential theory and Lefschetz theorems for arithmetic surfaces (Annales scientifiques de l'École Normale Supérieure, Sér. 4, 32 no. 2 (1999), p. 241-312) where this argument is explained for surfaces and adapted for arithmetic
surfaces.