As I understand it, this result is about algorithms that solve any instance of a given computational problem (like finding the shortest paths through a graph.) The specific problem instance (the graph) is the input to the algorithm. The result shows how to take an algorithm and construct another algorithm that solves the same class of problems in less space, by transforming how the algorithm reads and writes memory. Now you have an algorithm that still solves any instance of that problem but uses only sqrt(t) space.

So, to answer your question more directly: no, you don't have to have solved a specific instance of the problem, but you have to already have an algorithm that solves it in general.