I don't have any first-hand experience with freighter travel but all the research I have done suggest it's mostly done through various agents. Trips from Europe to Asia are readily available but none from Romania that I could find. All the agencies sell more-or-less the same trips, on container ships from CMA-CGM, Hanjin or German shipping lines. Here are a few examples of such trips (not all of them provide a direct call to South Korea):
The ship you photographed is a bulk carrier and apparently only very few of them take passengers (I have heard about grain “tramp” travel – meaning you don't know the exact destination until shortly before departure – from the Great Lakes to Europe and iron ore ships to Brazil but nothing else). Unlike container ships, bulkers don't serve fixed lines or regularly return to some “home” port, making either point-to-point return trips or tramp trade between ever changing destinations, not necessarily connected to the ship's flag country or owner.
Constanța is a large port for dry and liquid bulk but a relatively small one when it comes to containers. Major container companies have a hub-and-spoke network, much like airlines or courier companies. For example, CMA-CGM invested in the development of Malta Freeport, which does mostly transshipment and is well connected to Europe and Asia. It does also serve Romania and the Black Sea but with a feeder line connecting it with the Mediterranean and Northern Europe (don't know if it takes passengers). It's therefore unlikely that you would find a ship going all the way from Romania to Korea.
As far as I can tell, Mangalia isn't even a commercial port of any importance but really a shipyard, making it even less likely to find trips starting there through the usual channels.