Were there any significant opposition to the UK going to war with Argentina within UK politics, and was it an unpopular war at it's beginning?
There was no mainstream political opposition to the Falklands Conflict (NB it technically wasn't a war, as neither side declared one).
The political right in the UK were in power, and the mainstream political left were opposed to the regime in Argentina (which along with it's pretty much universally appalling human rights record was very much opposed to trade unions, in a far more brutal and permanent way than the UK)
There were some obejctors, either from the extreme left (so politically motivated) or out of pacifism, most notably:
The Campaign for Nuclear Disarmament (CND)
The Communist Party of Great Britain
The National Union of Students (NUS)
Pax Christi (Roman Catholic pacifist movement)
The Ecology Party (forerunner to the Green Party)
The Trotskyist Revolutionary Communist Party
There were a handful of Labour MPs who were privately (or very occasionally openly) opposed at the time, and a handful more who became opposed retrospectively, but to all effects and purposes there wasn't any meaningful opposition to the war.