I've researched the question quite a bit, might as well answer it.

First off, primary accounts from the time seem to show that the expectation was of a British victory.

I believe there are three main reasons the British were able to win despite the distance between the British isles and the Falkland Islands.

British power

Britain, despite it not being a superpower, was still a very strong nation. The British navy was considered to be one of the best in NATO and the world. During previous NATO excersizes the British navy outperformed most other navies. Meanwhile, budget cuts that were supposed to hurt the navy were not yet carried out.

Britain was also able to use it economic power to assist the military. The British, realizing that they must act fast in order not to allow Argentina controlling the islands to become the status quo, quickly mobilized its civilian navy, renting ships and yachts from the private sector in order to bolster its naval capacity. These civilian ships were used to carry fuel, supplies, and soldiers to the Falklands. Some of the were used as floating hospitals for wounded troops.

British troops were, on the whole, better trained and better supplied than the Argentinian forces. The Argentine military consisted of a mixed force of regular and conscripted troops. These conscripts were prone to disorder and fights between conscripts and their commanders were common.

On the other hand, British forces had use of SAS and Gurkha troops. These special forces had no problem with deployments far from home in tough conditions.

Another way Britain managed to cut down the distance between itself and the Falklands was through the use of Ascention island, the small island allowed Britain to cut down the distance between its closest base to the Falklands by half.

International pressure

With Frances help, Britain was able to pass a UNSC resolution condemning Argentina. This was helped by the USSR not vetoing the resolution. As a result of this, Britain was able to galvanize the western world to its aid. European and Commonwealth countries were quick to sanction Argentina, denying their economy funds and replacement weapons and ammunition.

Later on, when negotiating with Argentina proved futile, the US also joing in assisting the UK and sanctioning Argentina.

France assisted Britain even more. In the past, France had sold Argentina weapons, including missiles and planes. France sent sending air crews with the same model planes in order to train the British to fight them more effectively. France also passed information on any system Argentina had purchased from them.

Chile

Chile was in the middle of its own territorial dispute with Argentina. As a result, it believed that if Argentina won the Falklands war, it would be Argentinas next target.

Because of this, Chile provided the UK with covert support which was very important to Britain. This included delaying the handover of shipping vessals it had purchased from Britain, returning them fully fueled.

Chile allowed Britain to use it airspace, allowing Britain to utilize the SAS in ways it otherwise could not.

Chile also continually passed on intelligence to Britain, allowing them to sink the cruiser Belegrano, for example.

However, the biggest support Chile offered was not taking Argentinas side. Due to the uncertainty this instilled in the Argentine Junta, we'll trained troops who would have fought in the Falklands were instead stationed on the border with Chile.

Bibliography:

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Borders, W. (1982, March 31). Britons and Argentines Squaring Off. Retrieved from The New York Times: https://www.nytimes.com/1982/03/31/world/britons-and-argentines-squaring-off.html

Freedman, L. (1982, Fall). The War of the Falkland Islands, 1982. Foreign Affairs, 61(1), 196-210.

Freedman, L. (2005a). The Official History of the Falklands Campaign, Volume 1: The Origins of the Falklands War. London: Routledge, Taylor & Francis Group.

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