The surface currents would flow following the dominants winds. They would circle around the whole planet if there is no landmass to stop them. It looks like this except that, without the landmass, there is no curving arrows. It's just like the Antarctic circumpolar current.
The continent, starting from the pole would probably extend to the mid latitudes: 40-45. At this latitude, the surface currents and the dominant winds are normally going eastward. I said normally because such a large landmass will cause some perturbations.
The large continent will become very cold in winter. Probably as cold as Antarctica is during winter. Temperature below -50 and even below -60 Celsius would be very common over large areas. Looking at eastern Asia during summer, the temperature could be close to -10 or -15 near the coasts, close to 40 degrees of latitudes.
The interior would be excessively dry during winter. The coasts could receive some rain but again, by looking at rainfalls in Eastern Asia, it would not be a lot.
The interior of the continent become much hotter than in winter but large parts would probably remain under the freezing point. Taking Antarctica as an example, the 0 degree isotherm is close to the polar circle. Meaning that the areas under it have an average temperature under 0 even in their hottest month. Closer to the coast, temperatures could reach 15 degrees making it possible to practice agriculture.
The coastline would receive a good deal of rainfall since it would be located under the unstable polar front but the still cold temperatures from the interior will not pull the moisture toward the interior. Precipitations would be similar to those of Noway : over 1000 mm per year but would gradually diminish as we move inland. The precipitation on the pole would quite likely be close to those of Vostock : 22 mm per year. That info comes from Wikipedia but I think it's likely to be much lower than that.