I thought I already responded to this, but I'll try again since my answer doesn't seem to have posted.
Honestly, I think you're going to have trouble finding exactly what you want among Korean dramas. As AH pointed out there are many dramas where the prince hides his own identity, however a prince who is unaware of his identity is rare. 100 Days My Prince might be the closest (I haven't watched it, though).
The closest I can think of:
Queen Seondeok - I think this is almost exactly what you're looking for, however FL is the princess, not ML. She was sent out of the palace as a baby due to some prophecy. In one of the many (not very plausible) embellishments this drama makes to her life story, she grows up on the Silk Road, but she eventually returns, discovers her true identity, and (as the title suggests) becomes queen. I think there are a number of dramas with these princess-cast-out-of-the-palace-due-to-her-cursed-birth plot lines (though most don't become queens), if you are interested in that.
Arthdal Chronicles - This is set in the late Bronze Age in a society that does not yet have a monarchy, rather is a confederation of chiefdoms. Thus, there are no kings and princes. However, ML does have important parents who he is unaware of. He's also the reincarnation (?) of a god and only half human. So basically he has a pretty epic birth story (this drama takes inspiration from ancient foundation myths, so they are creating his character with the attributes commonly found in legendary kings). Though he is not really a hidden prince, he is destined to become a great leader (and he doesn't know any of this).
A couple comments about what other people recommended:
The Royal Gambler - I don't remember this drama very well, and I never finished it. But I think ML is the half-brother of the prince. I'm not sure he is actually a prince himself. I think they have the same mother? Or maybe his father is a mystery? Well maybe there is something about him becoming a prince in the episodes I didn't watch.
Hwarang - Major spoiler (kinda). Though it is true that late in the drama ML learns he has royal parentage, he never becomes a prince or king. Part of the reason this whole plot line is completely stupid is because SML is a famous historical figure, so the audience knows ML will not dethrone him. It's also stupid because the writers want you to believe that ML is a legitimate heir to the throne, but if you know some Silla history, you'll be skeptical of that. This is also a drama where the king hides his identity. He knows he's king, but others don't (and unlike many other instances of this trope, his identity is hidden for most of the drama).