How can African countries cope and flourish with today's fast globalizing world?

Obviously, Africa is a continent containing many different countries at different stages in their development, all facing different specific economic and cultural issues. What is generally true of most of sub-Saharan Africa is that it is part of the developing world. This answer will focus on issues current in 2016; conditions may change in the future.
For North Africa, different countries face different issues. For Libya, the main issue seems to be creating a robust government with a rule of law, as international trade is not possible with a single legitimate government and some degree of stability, including ways of guaranteeing the safety of foreign visitors and their property. Tunisia and Morocco are comparatively well-governed and face issues of building industries that can take advantage of their demographic dividends. Egypt faces political upheavals, declining GDP, and the challenge of political uncertainty, with an extremely repressive government being inherently unstable.
Several countries in Africa are considered failed or fragile states in which government is either non-existent (i.e. consisting of competing bands of warlords or lacking central control). In states such as Sudan, South Sudan, Yemen, Central African Republic, Chad, and Democratic Republic of the Congo, the first and most important task facing the nations is establishing a stable government and rule of law. Without these, becoming part of global trade networks and gaining the benefits of modern society are not possible.
In traditional resource-exporting nations in Africa, prospering in a globalized economy requires developing non-resource based industries, making them less vulnerable to fluctuations in commodity prices and raising GDP.
For many countries, increasing the ease of doing business and improving infrastructure (especially transport) will increase their ability to participate in global industries and provide jobs for young people.