I don't have complete confidence in this answer, but: Table $3$ in Fletcher lists parameters $(a_0, a_1, a_2, a_3, a_4)$ such that a general hypersurface of degree $1+\sum a_i$ in the weighted projective space $\mathbb{P}(a_0, a_1, a_2, a_3, a_4)$ is canonically embedded and has at worst terminal isolated quotient singularities. Let $H$ be such a hypersurface and let $\pi: X \to H$ be a resolution of singularities of such a weighted hypersurface.

Then I believe that

(1) $X$ is general type.

(2) $\chi(\mathcal{O}_X) = 1-\#(i : a_i=1)$.

(3) $X$ is simply connected.

In particular, if we take a quintuple with $a_0=1$ and $a_1$, $a_2$, $a_3$, $a_4 > 1$, we get a $3$-fold with holomorphic Euler characteristic zero. Examples in Fletcher's list are $(1,2,2,3,3)$, $(1,2,3,3,5)$, $(1,2,3,4,5)$, $(1,2,2,3,9)$, $(1,3,4,5,7)$, $(1,2,3,4,11)$ and $(1,3,4,5,14)$.

Regarding (1): Indeed, in these examples, $H$ is canonically embedded, so $\omega = \mathcal{O}(1)$. I am a little nervous about whether I am using language correctly for a singular variety, but I believe that I am.

Regarding (2): The resolution $\pi : X \to H$ has $\pi_{\ast} \mathcal{O}_X = \mathcal{O}_H$ and $R^j \pi_{\ast} \mathcal{O}_X$ for $j>0$, so we can compute $H^j(H, \mathcal{O})$ instead. I did this two ways. From Theorem 7.2 in Flecther, $H^{00}(H) = \mathbb{C}$, $H^{01}(H) = H^{02}(H) = 0$ and $H^{03}(H)$ is the degree $1$ piece of the graded ring, generated by $x_0$, $x_1$, $x_2$, $x_3$, $x_4$ in degrees $(a_0, a_1, a_2, a_3, a_4)$, modulo a Jacobian ideal of relations. But the lowest degree relation in the Jacobian ideal is $\left( \sum a_i + 1 \right) - \max(a_i) > 1$, so we are just counting the number of generators in degree $1$, which is $\#(i : a_i=1)$.

I wasn't confident that I had used Theorem 7.2 correctly, so I also directly computed the Hilbert series. Put $s=\sum a_i$. The Hilbert series is $$\frac{1-x^{s+1}}{\prod(1-x^{a_i})} = \frac{1-x^{s+1}}{\cdots+\#(i:a_i=1)x^{s-1}-x^s} = x + \#(i:a_i=1) + r(x)$$ where $r(x) \to 0$ as $x \to \infty$. When a rational function has $r(x) \to 0$ as $x \to \infty$ and denominator of the form $\prod (1-x^{a_i})$, then the coefficient of $x^n$ is quasi-polynomial in $n$ for $n$ all the way down to $0$. So we can compute $\chi(\mathcal{O}_H)$ as $$\lim_{x \to 0} r(x) = \lim_{x \to 0} \left( \frac{1-x^{s+1}}{\prod(1-x^{a_i})} - x - \#(i:a_i=1) \right) = 1-\#(i:a_i=1).$$

Regarding (3) Here is where I am a little nervous. Resolving an isolated quotient singularity shouldn't change $\pi_1$, so $\pi_1(X) = \pi_1(H)$. I want to say that I can apply the Lefschetz hyperplane theorem to the intersection of $\mathbb{P}(a_0, a_1, a_2, a_3, a_4)$ with a hyperplane in the appropriate embedding. $\mathbb{P}(a_0, a_1, a_2, a_3, a_4)$ is definitely simply connected, because all complete toric varieties are simply connected. (See EG Section 3.2 in Fulton's Toric Varieties.)

But I couldn't quickly find a reference for the Lefschetz hyperplane theorem for $\pi_1$ for singular varieties which says what I want it to say. I think that this is how it works, but I'm not an expert, so I'll leave the answer with this caveat.

Thanks to Chen Jiang's answer for pointing me to Fletcher's table.

I don't have complete confidence in this answer, but: Table $3$ in Iano-Fletcher lists parameters $(a_0, a_1, a_2, a_3, a_4)$ such that a general hypersurface of degree $1+\sum a_i$ in the weighted projective space $\mathbb{P}(a_0, a_1, a_2, a_3, a_4)$ is canonically embedded and has at worst terminal isolated quotient singularities. Let $H$ be such a hypersurface and let $\pi: X \to H$ be a resolution of singularities of such a weighted hypersurface.

Then I believe that

(1) $X$ is general type.

(2) $\chi(\mathcal{O}_X) = 1-\#(i : a_i=1)$.

(3) $X$ is simply connected.

In particular, if we take a quintuple with $a_0=1$ and $a_1$, $a_2$, $a_3$, $a_4 > 1$, we get a $3$-fold with holomorphic Euler characteristic zero. Examples in Fletcher's list are $(1,2,2,3,3)$, $(1,2,3,3,5)$, $(1,2,3,4,5)$, $(1,2,2,3,9)$, $(1,3,4,5,7)$, $(1,2,3,4,11)$ and $(1,3,4,5,14)$.

Regarding (1): Indeed, in these examples, $H$ is canonically embedded, so $\omega = \mathcal{O}(1)$. I am a little nervous about whether I am using language correctly for a singular variety, but I believe that I am.

Regarding (2): The resolution $\pi : X \to H$ has $\pi_{\ast} \mathcal{O}_X = \mathcal{O}_H$ and $R^j \pi_{\ast} \mathcal{O}_X$ for $j>0$, so we can compute $H^j(H, \mathcal{O})$ instead. I did this two ways. From Theorem 7.2 in Flecther, $H^{00}(H) = \mathbb{C}$, $H^{01}(H) = H^{02}(H) = 0$ and $H^{03}(H)$ is the degree $1$ piece of the graded ring, generated by $x_0$, $x_1$, $x_2$, $x_3$, $x_4$ in degrees $(a_0, a_1, a_2, a_3, a_4)$, modulo a Jacobian ideal of relations. But the lowest degree relation in the Jacobian ideal is $\left( \sum a_i + 1 \right) - \max(a_i) > 1$, so we are just counting the number of generators in degree $1$, which is $\#(i : a_i=1)$.

I wasn't confident that I had used Theorem 7.2 correctly, so I also directly computed the Hilbert series. Put $s=\sum a_i$. The Hilbert series is $$\frac{1-x^{s+1}}{\prod(1-x^{a_i})} = \frac{1-x^{s+1}}{\cdots+\#(i:a_i=1)x^{s-1}-x^s} = x + \#(i:a_i=1) + r(x)$$ where $r(x) \to 0$ as $x \to \infty$. When a rational function has $r(x) \to 0$ as $x \to \infty$ and denominator of the form $\prod (1-x^{a_i})$, then the coefficient of $x^n$ is quasi-polynomial in $n$ for $n$ all the way down to $0$. So we can compute $\chi(\mathcal{O}_H)$ as $$\lim_{x \to 0} r(x) = \lim_{x \to 0} \left( \frac{1-x^{s+1}}{\prod(1-x^{a_i})} - x - \#(i:a_i=1) \right) = 1-\#(i:a_i=1).$$

Regarding (3) Here is where I am a little nervous. Resolving an isolated quotient singularity shouldn't change $\pi_1$, so $\pi_1(X) = \pi_1(H)$. I want to say that I can apply the Lefschetz hyperplane theorem to the intersection of $\mathbb{P}(a_0, a_1, a_2, a_3, a_4)$ with a hyperplane in the appropriate embedding. $\mathbb{P}(a_0, a_1, a_2, a_3, a_4)$ is definitely simply connected, because all complete toric varieties are simply connected. (See EG Section 3.2 in Fulton's Toric Varieties.)

But I couldn't quickly find a reference for the Lefschetz hyperplane theorem for $\pi_1$ for singular varieties which says what I want it to say. I think that this is how it works, but I'm not an expert, so I'll leave the answer with this caveat.

Thanks to Chen Jiang's answer for pointing me to Iano-Fletcher's table.