Systematic approach or algorithm for designing a pushdown automata that accepts a langauge
Your question is something very natural, that comes up a lot when first encountering computational model.
The main problem is that your question does not "compile":

You ask if there is an algorithm that constructs PDAs. What would be the input for such an algorithm? This is the main crux of the problem.

Until you specify that, you won't really have an answer. As Yuval commented, if you get a CFG as input, then there is an easy procedure to convert it to a PDA.

Let's look at the other extreme: suppose you are given a Turing Machine, and you just ask whether there exists a PDA for its language (even without constructing it). As it turns out, this problem is undecidable (it's an immediate consequence of Rice's theorem, but in fact it's neither in RE nor in coRE).

This problem is something that often comes up in designing computational models - how do we specify what we want? If we give a rich specification formalism, such as Turing machines, then things become undecidable. And if we give something too poor, such as CFGs, then the construction task becomes easy, but writing down the specification becomes the hard part, so it's debatable whether we actually did any good...