Dublin is the capital of a pro-Western, NATO supporting democracy. The Soviet Union has tens of thousands of nuclear warheads (bear in mind that some UK cities would be allocated dozens of weapons); such a target more than merits one. The 'logic' of strategic nuclear war dictates that one must not only destroy current threats, but probable future threats such as a very capitalist and very Christian democracy such as the Republic of Ireland.
There's no real way that one can make the whole UK Government and the armed forces 'go under'; the very nature of civil defence planning means that the entire structure of both organisations is devolved and regionalised, to say nothing of being dispersed to reasonably safe locations pre-attack - unless the USSR bomb every inch of countryside and woods in the country they aren't going to wipe out the forces that have been basically hidden there.
Yes, Belfast will probably be hit; the Regional Government was to be based in Armagh, however, with the Army elsewhere. They will no doubt take a battering, but planning for the maintenance of command and control had been serious for some time.
The IRA might decide to get brave (or rather, stupid) and start causing trouble. Bear in mind, however, that at their height there were only 100-200 active 'soldiers' in the organisation, and only a fraction of them had heavy weapons. Even allowing for some influx of support post-attack, they are still going to be heavily outnumbered by even a decimated surviving Army, to say nothing of the Unionist Paramilitaries. Bear in mind that, even at their most brutal, the British during the Troubles were fighting with one and a half hands tied behind their back. There would be no such restraint displayed in the post-war world, and anybody idiotic enough to try and stir things up would be hunted down and killed outright like the animals that they, quite frankly, are.
The Irish Government are not stupid either; apart from the fact that their country will be starving and isolated, they know that they still can't win against the UK. It's not even in their best interests to - for all their differences the two nations will do much better to help each other post-strike rather than fight.
Should the Republic have some massive brainstorm and attack the North, it doesn't take a lot for the British to stop them. We don't need an aircraft carrier; a couple of Jaguars and a long-ish stretch of motorway on the mainland and a couple of small nuclear weapons would be enough to repulse any Irish intruders fairly effectively. Should the battered and angry British continuous governments be feeling particularly merciful or have no bombs available, conventional attacks from the mainland or from the sea, combined with the movement (difficult but possible) of reserves over to the Provinces would convincingly destroy the Irish Defence Forces outright.
The conscription idea doesn't really go anywhere; even in peacetime the Irish Army wouldn't be able to train, arm or feed this sort of influx in manpower. If they really, really wanted to give their scarecrow populace some sticks and order them against British tanks then they probably good, but most of the conscripts would probably decide to save time on the walking and just get shot for desertion there and then.