Well the way you are phrasing your question seems to indicate a misunderstanding.

It's not a matter of "turning a flat map into a projection" (to paraphrase you) It's a matter of converting a map from one projection to another.

A projection is a way of turning the 3D points of a planet's surface into a flat map. In mathematics it's something we call a 'function'. And each projection is a different function that flattens the points onto the map in a different way. So we don't say a map 'is a projection' but rather it 'has a projection' which is a description of how it relates to the surface of the Earth (or whatever else it' might be a map of)

Every projection of a curved planetary surface causes distortion though the particular distortion varies. In particular, no such projection preserves distance (Some preserve some distances, but none preserve all distances.) Even a map that's lumpy and distorted has a lumpy and distorted projection function, though we don't know what that function is.

So whatever your 'flat map' is, it has a projection. Assuming you know what that projection is, you can convert it to other projections using special software ranging from one off little utilities, to full on GIS Suites.

But if you just want to know the projection and get the technical details right, and don't care about changing it to other projections, you just need to pick one that suits the map (Of course now I'm assuming you are drawing a fictional map from scratch now and don't have any other maps or data you need to stay consistent with). It's best if you do this before you start designing it but you can pick one to make it work after the fact, at least usually.

So before going into detail, which of these are you after, if either?