For the sake of clarity/ease of understanding I am going to separate Lee Gon into two people the Young One and Lee Min Ho (LMH). The drama starts with a young LMH being betrayed by his uncle. He grows up and discovers the secret of the flute, goes through the inter-dimensional gate and ends up in Tae Eul's universe. They meet and discover the workings of the bad guy and in the process of trying to sort it out and stop/rectify things they fall in love. A whole bunch of things happen, then the not Tae Eul (Luna) from LMH's universe is sent across. She manages to poison LMH sending him to a clinic where he hears the flute play.
Now remember the black jacket with the hood Tae Eul gave LMH? He sees it and it sparks a memory in him. He works out that he was the one that saved himself from his uncle (the bad guy). When LMH wakes up in the clinic after hearing the flute he realises that the time has come to go back in time and save himself. NOTE: only when the flute is whole can it operate in space and time. When it is cut in half it can only travel across the dimensions.
So LMH and bad dude go through their respective gates (different b/c they have different halves). When they meet in the middle, though they cannot see each other, being on their half's path, they know the other is there. Because they are both there the flute is whole (it is in the same place & time) so they race to the other gate and take themselves back to the night of the treason.
LMH saves the Young One from dying, recognises his cousin/uncle (sorry, forgot their exact relationship), shoots him in the knee/leg and races back to the gate thinking he can get back to Tae Eul. It is only then does LMH realize he needs both halves of the flute to go through time. He reckons it will take 4 months inside the inter-dimensional world to get to the time/date he left Tae Eul. He makes the odd trip out of the gates to keep up with time (gotta know when he is or he'll miss it). I guess he figures he can also stop his Tae Eul universe self from being murdered so he calls the cops and tells them about it, unfortunately too late.
So every now and then he goes out to check on the date and leaves messages for Tae Eul to tell her he is making his way to her, via the slow route. He meets her asa kid after her mother's death, leaves a message for her at the phone booth and so on.
Every time you see LMH in the black jacket with the hood (in Ep 14) it is the current Lee Gon. You only ever see the current Lee Gon whether it be in present day or when he goes back in time to fix things. You will never see him otherwise. The only exception to this is when, after he has fixed things (ish) Tae Eul meets the guy that looks like LMH (in her universe) as bad guy wasn't able to murder him when he was a kid.
That was a long winded way to explain that his current self met Court Lady Noh in the past on the night of the treason. Have you finished the series yet? I don't want to explain more in case of spoilers if you haven't.
Hope that makes sense.