Since Macbeth ends with the death of its titular character, we're unable to say for sure exactly how Malcolm can or will face Scotland's undoubtedly numerous problems, and thoughts on this question will be pretty speculative. However, based on what we know of Malcolm, and based on what we know of Scotland's post-Macbeth woes, it's possible to make some guesses.
First, let's consider the actual problems Scotland faces: as the play progresses, Macbeth becomes an increasingly paranoid tyrant, killing allies (and, at certain times, his allies' families) in order to consolidate power. Thus, Scotland will face some power vacuums brought on by the collapse of prominent aristocratic families. Moreover, the country as a whole is likely to be wary and distrustful of any kind of government for quite some time; after all, why should citizens of Scotland trust a new monarchy if the last one abused them in order to gain power?
Now, let's consider Malcolm: a cautious and clever individual, Malcolm is smart enough to leave Scotland after his father's death, and to return again to orchestrate Macbeth's downfall. Also, in a somewhat calculating way, Malcolm motivates Macduff to use the death of his family as motivation to seek revenge against Macbeth. Furthermore, at the end of the play, Malcolm presents us with a depiction of dignified royalty, promising to restore peace and order to Scotland. As such, it seems as though Malcolm not only knows how to play the political game, but also how to instill hope in the hearts of his subjects, and his intentions seem to be mostly noble. In that case, though we can't be sure exactly how the country's fortunes will play out after Macbeth's demise, we can hypothesize that Malcolm will use his nobility and political intelligence to restore some stability and confidence to Scotland.