We know by some facts from Kobayashi, if the Kahler manifold $M$ has positive first Chern class, i.e., $c_1 (M)>0$ then $M$ is simply connected. So if $c_1 (M)<0$ under which assumption on $M$ , we have $π_1(M)={e}$.
This is really more of an extended comment, since I'm not sure how to give a definite answer. When $\dim M=1$, $c_1(M)>0$ if and only if $M= \mathbb{C}\mathbb{P}^1$ if and only if $M$ is simply connected, by the uniformization theorem. In the dimension $2$, things are more complicated. Certainly simply connected with $c_1 < 0$ exist. For example, any surface in $\mathbb{C}\mathbb{P}^3$ of degree $5$ or more will work. However, there are also plenty of nonsimply connected examples (products of curves of large genus, ball quotients...). Does this help? Added Explanation: To explain where the examples are coming from and to answer your 2nd comment, let me explain I'm using Kodaira's embedding theorem to translate $c_1(M)<0$ to ampleness of the canonical bundle $K$. This condition is stable under products. 

