I believe that you may be asking slightly the wrong question. You ask:
"Will Ships still be useful on a super-continent"
To which my answer is .... Yes!
But a slightly better question is:
"Where will trade occur, and through what means"
Curiously, predicting trade is often a fairly good proxy for technical prowess and societal development (ie trade centers are often among the first places to adopt systems of currency, economics and written languages).
So, the societal hotspots for your continent may be:
The area around Solnia is crammed with countries and easy land routes, quickly supplemented by sea trade. The sea is sheltered by the spit of Uruslania, and so sea travel is practical and easy in this region. As soon as rudimentary boats have been created, Velsia will be able to trade "across the sea" with dozens of countries. (I can imagine Velsa and Riveso as the capital of the world in the early days of sea travel - similar to what Rome was. Heck, put a shoe-shape on the end of the Riveso Peninsula and it Velsa and Riveso are practically Italy and Greece)
Similar stories play out in the other circled regions, though probably to a lesser extend. Crinteto region I imagine as a bit of a late-comer to the party, as it is more exposed. Ship designs from that region have smaller cargo payloads, but their shipbuilding is highly advanced. Their captains known for sea-faring skill. But they don't have a great economy.
These hotspots will generate internal trade via short sea routes, but until shipping advances allow longer distance travel, they won't really interact other than via long-distance-road transport - silk road style. I don't know exactly what tech level required is. 17th century ships totally could, but even in the 2nd century, India was trading with China via the sea. Anyway, after ships reach some level of seaworthyness able to cross long distances (perhaps the designs are originally Crinteto?), sea trade really takes off, and the land routes change.
In particular, notice how narrow it is between Torsand and Freinsia? That's a prime land transport route. Goods may be transported in from the south, hit a port at Freinsia, be transported by land to Torsand, and from their access dozens of other countries. Iragoa or Numidia may frequently fight for trade deals, or may become neutral (like Switzerland, which historically had a lot of trade passing through).
Mariteto and Cirta are also a candidate for a great overland route, but there aren't many socieities over on the west. But imagine if Verisianio wanted to trade with Parasinia. Either they have to go anticlockwise and possibly use the overland route at Freinsia, or they go anticlockwise and they cut a long way off the journey by overlanding through Mariteto. However, Mariteto does have something else going for it: It'd be a great canal building project. I can imagine distant wealthy societies paying for the cutting of a canal through places like Mariteto and Cirintia. Virdria may be a bit remote to be economically feasible to drive a canal through.
Here's a rough trade map showing some of the prime trade routes that I can forsee arising. Of course I don't know what the mountains are like, or if one country is mostly swamp, but it may give you some ideas where to look for interesting stories.
Some things to note:
- Ships could stop off anywhere along the coast, not only at the points I've indicated. The points I've indicated them stopping off at are what I consider likely travel hubs.
- The Dorn empire may make for easy land trading as fewer political boundaries occur. Similar reasons are for the routing of land routes through as few-er countries as possible.
Oh, and vikings are totally found in Iberia and raid across to Parasina.
A couple interesting things from comments:
- @FuzzyChef points out the interior of supercontinents are often deserts - which are inhospitable and hard to travel across. This will help drive trade via sea, and may also explain why the west doesn't have many empires.
- @clockw0rk mentions that the underwater cities will also drive more sea trade - both because they provide trade destinations and because they may be able to with assist technology and personal. I do however think that most transport would be open sea rather than through underwater caves - though underwater caves would definitely be the cool solution.