Organizing an anarchist society on a global highly interconnected scale isn't all that different from doing it on a smaller scale. One has worker assemblies, community assemblies, etc. and they all coordinate with each other by forming confederations using spokescouncils. There would probably be regional confederations, and confederations of those confederations for issues that extend beyond that region and so on up. This can be applied on any scale, from local to global to intergalatic.
Of course, the kind of global connections we have now probably would probably be dismantled because they were designed to profit the rich & powerful, not the average person. For example, a company can often save a lot on labor costs by setting up sweatshops in a different country with a weaker labor movement and/or more business-friendly labor laws. This wastes resources because the goods produced now have to be shipped from the (far away) point(s) of production to the location(s) where they are consumed. The capitalist gains from this because the lower labor costs more than offset the shipping costs but it is not an efficient use of resources. Where possible, it is better to produce things in the same area as where it will be consumed because it doesn't have to be shipped as far. This isn't always the case - certain places can't grow certain crops and so on - but it doesn't need to be taken to the extreme we have today.
The transition from the present society to an anarchist one is a little trickier. An anarchist revolution that is isolated in a single country would be in a difficult position, but not necessarily an impossible one. There are economic issues as well as political-military ones. If the capitalist nations are willing, trade could continue with those nations - worker collectives can export products to capitalist nations and import products produced by corporations. The terms of trade would probably change, since imperialism in that country would have been expelled, but hypothetically it could continue after the revolution.
However, the remaining nation-states would probably be extremely hostile towards the anarchist society and would place sanctions on it. Production would have to be reoriented towards domestic production. To an extent, this is something that should be done anyway, for reasons outlined above, but it takes on an added urgency in the case of an isolated anarchist society. The various worker assemblies (and federations of them, etc.) can convert workplaces and other economic assets away from export and towards whatever domestic needs were formerly fulfilled by imports. This can also be done to get rid of things like advertising, which serve no useful purpose, and to eliminate other capitalist inefficiencies. The Spanish revolution provides a precedent for anarchist reorganization of production, with its quick switch to wartime production by the self-managed collectives. Much of this depends on the characteristics of the country which underwent an anarchist revolution. If it has lots of rescources this will be made easier, if it is lacking something then they may have to find a substitute (or go without) at least until the revolution spreads.
A bigger threat is the possibility of military intervention to crush the revolution. Even if there aren't any investments in the country, the capitalist states would tend to attack the anarchist society because it presents a "threat of a good example." People in other countries will see the example of anarchy in that society and may think it a good idea & attempt to imitate it in their own country. It could inspire revolt in other countries, which will drive capitalist states to attack the revolution, to prevent that "threat of a good example."
Such reactions to revolutions are not unique to anarchy or to our current era of "globalization." After the Russian revolution the United States, England and Japan invaded Russia to destroy the Soviet government. The European Monarchies used military forced and eventually formed a giant alliance to suppress the republican revolutionaries in the late 18th and early 19th centuries. The United States has attacked every attempt at socialist revolution. There will probably be similar attacks on anarchist revolutions.
These attacks could be dealt with in the same manner as defeating domestic counter-revolutionary armies (arm the people, etc). The more isolated the revolution is, the more successful these military attacks will probably be. If the revolution can be spread it will undermine the position of the imperialists and make it more difficult for them to suppress it. It's much easier to suppress a revolution that is limited to a small island than one that spans an entire continent. The promotion of revolutionary movements in the imperialist countries can also be used to defend the revolution by getting soldiers to mutiny, workers to strike against and so on. Each country the revolution spreads to is another country that won't be invading the countries which have already joined the revolution. The defeat of the imperialist armies may very well be possible if the revolution stays isolated (there are many examples of guerilla armies defeating invaders with superior firepower), but it will be easier if it spreads.
A revolution isolated in the United States would have a much easier time than a revolution isolated elsewhere. You wouldn't have to worry about a military attack and the US has many natural resources. If the rest of the world put sanctions on the US their economy would collaps. A revolution in the US would probably trigger the collapse of global capitalism and a number of revolutions. Many governments are dependant on US support to stay in power, without them they would fall. Anarchist ideology, having brought about the fall of an empire most of the world hates, would probably spread like wildfire. Within a couple years there would probaby be more anarchist revolutions in other parts of the world and it would spread to voer most of the globe fairly quickly because the collapse of the empire would make it much more difficult for the remaining capitalists to prevent it. The US is and has been the #1 counter-revolutionary force for some time now, without it world politics would change dramatically. This is unlikely, though, because the US anarchist movement is weak compared to many outside the US (such as those in Latin America & Europe), among other reasons.
Advocacy of world revolution has a long tradition in anarchist circles. The greater the area an anarchist revolution encompases the better the odds it will defeat attempts at counter-revolution. The more widespread it is the less economic difficulties it will encounter (it will have greater access to resources and will not have to reorient the economy as quickly). An isolated revolution isn't necessarily doomed, but it will have an easier time the more rapidly the revolution spreads.
There are many precedents for simultanious revolutions on an international scale. The 1848 revolutions are a classic example, as is the 1917-1922 period of world unrest. The fall of the Soviet empire saw numerous revolutions all happening at around the same time. It is possible that the fall of the American empire will see a similar phenomenon. Even when revolutions aren't stacked as closely as those examples, revolutions often tend to come in waves. There's the waves of republican revolutions initiated by the American, Fench & Haitian revolutions, the wave of "third world" revolutions after WW2 and others.
Today's 'globalization' probably makes this even more likely because things are even more interconnected. What happens in one area has a profound impact in another. Many soial movements today are more international than ever before. The simultanious demonstrations against the Iraq war (such as F15) and what American media calls the "anti-globalization" movement are two recent examples.