The classic example of an almost complex manifold that is not a complex manifold is the six-sphere $S^6$. Consider $S^6$ in $\mathbb{R}^7 = \mathrm{im}\,\mathbb{O}$ as the set of unit norm imaginary octonions. The almost complex structure on $S^6$ is defined by $J_p v = p \times v$, where $p\in S^6$ and $v\in T_p S^6$ and $\times$ stands for the vector product on $\mathbb{R}^7$.
This almost complex structure cannot be induced by a complex atlas on $S^6$ because the Nijenhuis tensor $N_J$ doesn't vanish (cf. the Nirenberg Newlander theorem).
A result of Borel-Serre states that the only spheres endowed with an almost complex structure are $S^2$ and $S^6$. (The one on $S^2$ is a complex structure.) Up to now, it is not known whether there exists a complex structure on $S^6$.
As a reference I mention the Wikipedia page on almost complex manifolds.
In this sense, a complex structure is an endomorphism of the tangent bundle $J:TM\to TM$ such that $J^2=-1$ (it plays the role of the multiplication by $\sqrt{-1}$). Saying that a Riemannian metric $g$ preserves the complex structure amounts to saying that $g(Jv,Jv)=g(v,v)$ for all tangent vector $v$.
Note that this forces the (real) dimension of $M$ to be even, and that if $M$ is a complex manifold, taking holomorphic charts $(U,\varphi=(z^1,\dots,z^n)=(x^1,y^1,\dots,x^n,z^n))$ and defining locally a $J_0:TM\to TM$ by $J_0(\frac{\partial}{\partial x^i})=\frac{\partial}{\partial y^i}$ and $J_0(\frac{\partial}{\partial y^i})=-\frac{\partial}{\partial x^i}$ defines a canonical complex structure.
Moreover, given any $J$, the complexification of the (real) tangent bundle of $M$ is written as $TM\otimes\mathbb{C}:=T_\mathbb{C}M=T^{(1,0)}M\oplus T^{(0,1)}M$ where $T^{(1,0)}M$ is the vector bundle associated to the eigenvalue $i$ of $J_\mathbb{C}$ (where $J_\mathbb{C}(v\otimes z)=J(v)\otimes z$) and $T^{(0,1)}M$ is the vector bundle associated to the eigenvalue $-i$ of $J$. Thus you have a $\mathbb{C}$-linear isomorphism between $(TM,J)$ and $T^{(1,0)}M$ as complex bundles (and a $\mathbb{C}$-antilinear one between $(TM,J)$ and $T^{(0,1)}M$).
Best Answer
Yes, it is true, it is a corollary of the inverse function theorem in complex analysis. If it is a diffeomorphism, locally, its complex Jacobian is invertible and you can apply:
https://en.wikipedia.org/wiki/Inverse_function_theorem#Holomorphic_functions