Problem Statement and Solution Approach

This is a friendly nitpick so feel free to ignore...I try to avoid referencing Agile as a methodology since it's not something you do. It's the set of values and principles used to create the mindset that differs from Traditional mindset. Scrum for example is one Agile based framework/methodology and Waterfall is one Traditional based framework/methodology.

I think the answer you're looking for is the Epic. The story of stories.

Try to create a single story where a value statement is a transformation of the problem statement. Then write the acceptance criteria for this story. Those ac's should come out to the same as the solution approach, but if not, it would likely indicate the original solution approach needed improvement.

Then change the story into an epic, and create the necessary stories to meet of the epic's ac's.

Where do you mention Purpose of the project (Problem Statement) and Solution Approach in Agile methodology?

In Scrum, each Sprint can be seen as a project, the purpose of which is expressed as the Sprint Goal. Wider goals may be articulated in the Product Backlog.

The current solution approach will be forecast in the Sprint Backlog, and potentially in the Product Backlog for the longer term.