This is possible. Basically, if $N$ is a point on the boundary of the nef cone that is not in the span of the rational points on the boundary, then we can approximate $N$ arbitrarily closely by rational points in the interior. For this, we need a variety whose nef cone is not a rational polyhedral cone:
Example. Let $X = E \times E$, where $E$ is a non-CM elliptic curve. Then $\operatorname{NS}(X) = \mathbb Z f_1 \oplus \mathbb Z f_2 \oplus \mathbb Z \delta$, where $f_1 = [p \times E]$, $f_2 = [E \times p]$, and $\delta = [\Delta]$. Moreover, the intersection matrix with respect to the basis $(f_1,f_2,\delta)$ is given by
$$\begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix}.$$
Letting $H = f_1 + f_2 + \delta$, the nef cone is given by the inequalities $D^2 \geq 0$, $D \cdot H \geq 0$, i.e.
\begin{align*}
xy + yz + zx \geq 0,\\
x + y + z \geq 0.
\end{align*}
See for example Lazarsfeld's Positivity in algebraic geometry I, Lemma 1.5.4.
The key observation is that the boundary of the nef cone contains rays that are not defined over $\mathbb Q$. For example, it contains the point $N = (1,\sqrt{2},\sqrt{2}-2)$, which satisfies $N^2 = 0$.
We will use the following lemma.
Lemma. If $\alpha \in \mathbb R/\mathbb Z$ has infinite order, then the set $\mathbb N \alpha \subseteq \mathbb R/\mathbb Z$ is dense.
In other words, if $\alpha \in \mathbb R$ is irrational, then the fractional parts of $n\alpha$ for $n \in \mathbb N$ are dense in $[0,1]$. This is a strengthening of the Dirichlet approximation theorem.
Proof. This is well-known. It's sometimes stated for $\mathbb Z \alpha$ instead of $\mathbb N \alpha$ (e.g. here and here), but the proof can be easily modified. $\square$
Example (continued). Let $\varepsilon > 0$. Applying the lemma to the interval $(-\varepsilon,0) \pmod{\mathbb Z}$, we conclude that there exist $m,n \in \mathbb N$ such that $n - \varepsilon < m \sqrt{2} < n$, i.e.
$$0 < n - m\sqrt{2} < \varepsilon.$$
Consider the element $D = (m,n,n-2m)$; this is an approximation of $mN$. We compute
\begin{align*}
D^2 &= 2\bigg(mn + n(n-2m) + (n-2m)m\bigg) \\
&= 2\big(n^2 - 2m^2\big) > 0,
\end{align*}
and $D \cdot H = 2(2n - m) > 2(n - m\sqrt{2}) > 0$, so $D$ is in the interior of the nef cone, hence ample. Since $K = 0$ and $\chi(\mathcal O_X) = 0$ on an abelian surface, Riemann–Roch gives
$$h^0(D) - h^1(D) + h^0(-D) = \frac{1}{2} D^2.$$
This forces $h^0(D) > 0$, since $D^2$ is positive and $h^0(-D) = 0$. Thus, there exists an effective curve $C \in |D|$. Finally, we compute
\begin{align*}
C \cdot N &= \bigg(m\sqrt{2} + n\bigg) + \bigg(n(\sqrt{2}-2) + (n-2m)\sqrt{2}\bigg) + \bigg(m(\sqrt{2}-2)+(n-2m) \bigg) \\
&= 2n\sqrt{2} - 4m = 2\sqrt{2}\bigg(n-m\sqrt{2} \bigg) < 2\sqrt{2} \varepsilon.
\end{align*}
Since $\varepsilon$ is arbitrary, we conclude that there exist effective curves $C$ such that the product $C \cdot N$ is arbitrarily small. $\square$
