One way to approach both the general and the specialized questions you raise is to study these Lie groups as $\mathbb{R}$-points of complex semisimple algebraic groups defined over $\mathbb{R}$. In this framework there is a lot of relevant work on both structure and homomorphisms by Borel and Tits, following their foundational 1965 paper on reductive groups. Fundamental groups are here characterized algebraically in terms of the position of the character group of a maximal torus inside the full abstract group of weights for the given root system.

Probably the most relevant part of their work for your purpose appears in the "complements" to their 1965 paper: see Publ. Math. Inst. Hautes Etudes Sci. 41 (1972), especially Section 4 *Le groupe fondamental de certains sous-groupes semi-simples*. This should be available online via numdam.org. (They also study homomorphisms between algebraic groups over various fields of definition in their 1973 *Annals* paper, online via JSTOR.) It takes quite a bit of work to get into their notational world, but their results are probably as precise as any you will find in the Lie group literature. (On the other hand, not every semisimple Lie group is algebraic, so it depends on your actual set-up.)