It feels awkward to answer the question posed by myself, but I was told it's okay to do so. There are few points I want to say here.

First of all, regarding Remy's question. On a compact complex surface $X$, every holomorphic 1-form is closed, hence $h^0(\Omega_X) \leq h^1(O_X)$ holds true in this case (by David Speyer's answer). So suppose $\eta$ is a holomorphic 1-form on $X$, then $\int_X d\eta \wedge \bar{d\eta}=0$ by Stokes. Then one expands $d\eta \wedge \bar{d\eta}$ in local coordinate to see that this would imply $d\eta=0$. More generally on a compact complex n-fold, every holomorphic (n-1)-form is closed by the same argument.

Secondly, my main question is actually whether the inequality $h^0(\Omega_X) \leq h^1(O_X)$ holds in general. Now I knew a counterexample to this inequality which is Iwasawa manifold. One can just consult this and that. Probably I should describe Iwasawa manifold here in case someone forgot the definition. Let
$G=\{\begin{bmatrix}
1 &a & b \\
0 &1 & c \\
0 &0 & 0
\end{bmatrix}|a, b, c \in \mathbb{C}\}$
be the Heisenberg group of complex coefficient and let $\Gamma$ be the subgroup with entries in $\mathbb{Z}[i]$, then we form a quotient $X=\Gamma \backslash G$. By projecting to $(a,c)$ coordinates, we exhibit $X$ as a holomorphic fibration with fiber $E$ and base $E \times E$, where $E$ is the beautiful elliptic with j-invariant 1728. One compute directly that $\Gamma^{ab}=\mathbb{Z}[i]^2$, hence $h^1(X,\mathbb{C})=4$. One sees (arguably not so obvious) that $da$, $dc$, and $db-adc$ form a basis of $H^0(X,\Omega^1_X)$. Applying Leray spectral sequence to the projection $X \to E \times E$, one can actually show that $h^1(O_X)=2$ (the point being that the differential $d_2: H^0(R^1\pi_*(O_X)=O_{E \times E}) \to H^2(O_{E\times E})$ is nonzero, and it sends 1 to $\bar{da} \wedge \bar{dc}$ if one normalizes everything appropriately).

Thirdly, Hopf surface would also give an example where the inequality $h^{01} \geq z^{10}$ in Davide Speyer's answer is not an equality ($z^{10}=h^{10}=0$ in that case).

Lastly, I was also wondering if there's an example where the Hodge-deRham doesn't degenerate at $E_2$ page. I was informed by my friend yesterday that such examples (nilmanifold) do exist, but I know nothing about it so perhaps I should stop here.