It might be possible for Argentina to develop the capability to boost a payload to LEO satellite altitude in that time frame. It wouldn't have to achieve orbital velocity, just be able to get near the satellite before it fell back to earth.
However, one has to remember the reason Galtieri started the Falklands war: to distract the people's attention away from Argentina's dismal domestic economy.
So, while getting a payload into space, and attacking a satellite might be possible, Argentina did not have the economy to fund such an effort.
And if they did have the money to develop such a system, they probably wouldn't have started that war.
Also consider that most satellite data available to the UK, came from US satellites. Attacking them would have had far reaching consequences.
A more realistic scenario would be Argentina developing its own anti-ship missile technology. Far less expensive than ASAT, and far more effective for that particular conflict.
At the start of the war, Argentina only had a few Exocets, and shot almost all they had. If they had possessed more, and were able to take out the UK's two carriers, they could have defeated the UK invasion force, as the British would have had no air cover for their ships, or their troops on the ground.
That conflict was a lot closer than the outcome might suggest.