Without their bread basket, they'd have to import more food - on one hand that means earlier liberalization of their food market, which would be good for dealing with the US and Europe, on the other hand it would mean earlier and more intense efforts towards family planning, to avoid such a dependency.
All in all, "South Japan" looses some of the inhabitants of Hokkaido, some more people to the economic problems after WWII (starvation), and a lot more due to slower population growth afterwards. South Japan would by today have something like 80 million people, while North Japan would have about 5 Million people, many of them Russian or from other communist nations.
Japans position in Asia would be much weaker, but that might also mean less suspicion by formerly Japanese occupied countries.
Therefore, South Japan might be economically nearly as strong as Japan today, and well integrated into a free market and a common defense of other capitalist countries in the area.
On the other hand, there might be more of a revanchist thinking in Japan, maybe even similar to what brewed in Germany after WWI. If that is the case, Japan will be pretty unpopular not only in communist countries, but also in capitalist countries, especially in Asia. We'd have a diplomatically and economically much more isolated Asian Prussia there, ready to arm itself at the first opportunity - some in the Korean war, some in the Vietnam war, and so on.
A right-wing government with nuclear weapons might even try to seize the opportunity of the fall of the Soviet Union to grab Hokkaido and the Kuriles, and, if things go well, also Sachalin. An attack on Kamchatka is also started, but mainly for bargaining in the following peace talks.
It might even work, as the only valuable possession in this list is Hokkaido and the remains of the Soviet Union might think it's unrealistic to hold on to that. So no nuclear weapons are used by either side. International pressure might make the Japanese give up Sachalin in the peace talks.
Such a war would have a lot of consequences for Russia, though - difficult to speculate there. A lot of defense activity might even make the slump after the change to capitalism much less severe and much more excusable to the people.