"Any amount of" is often used in Ireland to mean "a large amount of".
This can be confusing to people only used to other forms of English, where the only figurative use of "any amount" is normally in the negative to refer to a small amount. However, it makes sense if you compare it to the idiom "any number of" used more widely to mean "a large number of" or "an arbitrarily large number of".
"…road under them" quite literally refers to the fact that road is generally underneath you; there is a lot of good-quality road in Sligo.
He's saying that the people of Sligo have a lot of good quality roads and that this is because McSharry has become a government minister and is using his influence to prioritise road-building in that area and hence win the favour of his constituents at the detriment of being fair to the rest of the country.
And sure enough, while I was only deducing that from the story there, I see that former minister Ray MacSharry was from Sligo. Since there was never a TD called McSharry and the only people called "MacSharry" listed in the Oireachtas database are him in the Dáil and his son in the Seanad, I would guess that he is referring to this real former minister, rather than a fictitious one. (I'm not sure I'd agree that the roads in Sligo are particularly good, but the story is set in Galway and Mayo and the roads there were much worse until quite recently.
So:
You would travel quickly through Sligo, normally, unless it was on Thursday, because there have been lots of good roads built and improved in Sligo since Minster MacSharry used his influence to favour that region.