Paul doesn't address the historicity of the earth didactically. Yes, in Rom 5:12-18 and 1 Cor 15:21,22 Paul references that the entire human race fell in Adam. But this doesn't state that Adam was the first and only man God created and Eve the only woman God created from whom the entire race came. Of course, the passage in Romans does state that Adam was the federal head of the entire race and which was negatively effected by his transgression, i.e., Original Sin; imputed guilt and inherited corruption of nature. The specifics of the creation are to be found in Genesis 1-3, but even there no mention is given as to the time it was created. The creation narrative does strongly state that God created the earth "mature" and thereafter Adam and Eve were created thus disallowing the Theistic Evolution view. Some, e.g., Archbishop Ussher used the chronologies of the OT to determine the age of the earth in his book The Annals of the Old Testament (1650).

Personally, methinks that the case can be made for a young earth although I would be hesitant to state an exact age of the earth's creation because I am not sure if the chronologies given in the OT are 100% complete. Whatever the exact date that God created the earth and Adam it surely wasn't 100,000+ years ago or some ridiculous period spanning billions or even millions of years ago as some prognosticate. Nor did the creation evolve from one micro organism as Evolution postulates.

Sooooo, the bottom line in regard to finding proof for a young earth in Paul I believe is an exercise in futility since I find nothing to support that venture. ALL Scripture is given by inspiration of God and therefore "proof" for any doctrine or view must be based upon the entire teaching of Scripture, aka: The Analogy of Faith. :