there were no OZ flights to Mongolia?
Unfortunately, as is, you don't have any access to a lounge based on status or cabin class.
As mentioned earlier, you'll have options if you have Priority Pass.
However, the big wrinkle, even if you can access a lounge, is that the earliest they open is 5am for the OZ J lounge. My understanding is that the SFO-ICN flight usually arrives quite a bit earlier than the scheduled arrival of 4am, meaning you've got upwards of 2 hours of dead time EVEN WITH lounge access. Making matters worse, most of the shops and food outlets in the airport will be closed too. I have heard there is a 24hour food outlet, but I think that's about it when it comes to things that are open...
My suggestion? Go to the free shower and take your time...