It depends on how you want your magic radiation to work, but here is one standpoint:
Radiation of any kind causes cellular damage by (among other things) breaking apart nucleotide sequences so they cannot be easily repaired (consequently, our cells are amazing at DNA repair, repairing as many as 1 million individual genetic lesions per cell ). Now, a normal level of damage is caused by all sorts of things, from environmental factors to regular oxidation due to metabolic activity, but each repair made to the nucleotides can introduce an imperfection or an error, the compounding of these errors eventually leads to diseases (such as cancer).
Large doses of radiation lead to extreme illness and death, and all sorts of screwy things happen to the body post-radiation exposure. There are stories of Hiroshima victims who had rapidly growing fingernails that had blood vessels in them, so when cut would bleed extensively. Typically the damage is most severe around bone marrow, GI tracts, immune system, and hair follicles. Given data on effects of radiation exposure to fetuses the easiest way to cause the effects in the world that you want is to have people dosed with 1-2 GY of radiation, which will impact fetuses (by miscarriage, or genetic malformation) but not cause mass deaths among the adults.
As to how your hero became superman, it was random chance. The DNA damage that occurred to him was minor enough to not cause him to be miscarried, and early enough to impact stem cells that would become his muscles. A random repair to his genes increased his strength, and was replicated through his whole body by the fact that this damage was done to his progenitor stem cells when he was an early fetus. (Note: low-doses of radiation are the best scenario here, as higher doses would likely just kill the fetus or deform him).
The harder part becomes testing for this. In this scenario, the fact that he was born healthy (and above average) was probably taken as a blessing, and since "all babies exposed to magic radiation die" nobody bothered testing him (if they even bothered developing a test). Likely the test of exposure was more a test of overall health than a scientific test to determine exposure. It is possible at some point in the future, some test is developed that can determine if someone was ever exposed to radiation, or determine that some part of his genetics was altered by exposure to radiation, but given he had been an adventurer/hero his whole life, it is likely he had exposure at some point anyway.
Now, to get to your point of magic causing some sign that it was a magical radiation exposure and not a "natural" genetic malformation, I think we can go into CRISPR territory. Perhaps, instead of being generic radiation, the magical radiation is "smart" radiation. Based on the kind of spell that created it, it does specific things to cells. Most of the time, the level of exposure causes so many changes to fetal tissue that the fetus is changed to become incompatible with life, and terminated by the mother's body, or the changes are so severe they cause rampant birth defects. As above, perhaps your hero was lucky and only got exposed a little, or perhaps the magic that hit him did so at a perfect time, where most of the effects were compatible with one another and did not deform him (save for his improved abilities). In any case, these changes can leave a certain genetic tagging in place, perhaps some sequence all magic imprints on genes that are changed (rewriting the largely useless junk sequences that are present in all DNA sequences left over from evolution). The reason it was never discovered is as above, your hero was never sick as a child so nobody thought it was possible he was magically afflicted (which raises the question of how many others there might be). When he finally gets around to getting tested, the truth is revealed.
This could also explain any other changes you want him to have (a lessened or inability to perform magic). Since this is magic radiation, there's no reason it can't be smart enough to make specific changes and leave tags in the genetics.