The paperwork is a big problem:
1.the carnet de passage will stump you I fear.
2 Registering in Ireland may be impossible.
1.Look at http://sidetour.free.fr/e_index.php
to see the problems they had doing something similar to what you propose. It really ruined things for them.
You would need, as far as I can imagine, a Russian carnet de passage. Eek. How's your Russian? Do you rally feel able to do all the Russian paperwork to take the bike out of Russia? Pay into a Russian bank, deal with Russian officials for weeks on end?
Perhaps Russia has a special vehicle export deal with the EU that I don't know about. You need to find this out beforehand.
2. Then you would have to formally import a Russian vehicle into Ireland. Not so hard maybe. But you will not be able to register it for the road: the sidecar is on the wrong side. It fails MOT in UK. I suspect that Ireland has similar rules. You need to find out beforehand.
(I am assuming you mean a Ural sidecar, not solo).
In UK some MOT stations don't know about this, and will register it anyway, and some will do it as a Trike, but both these dodges are, well, dodgy, and the Trike loophole has been noticed by the government in recent years so that they are tightening it.
You may end up only being able to ride up and down your driveway. But, as I say, getting it out of Russia will be a task in itself.
Sorry to be so negative. Understand that I am trying to save you from weeks of teeth gnashing on the Russian border.
Why not buy at home? Then everything is easy to put in place, and you can concentrate on having a good time. There is no need to make your life too hard. Owning a Ural has enough challenges on that score
Buying a local Ural and riding it to and from the factory in Irbit is very doable, and I am sure they would be pleased to see you.
My advice: don't give up on the dream--just ammend it a bit.