It is getting fairly wide spread these days.
My knowledge could be a little rusty now as I'm out of the city a long while now but ...
Temple bar general area would be a good spot during the day.
You have the likes of the Temple Bar bar which have a vast array of whiskeys
Oliver St John Gogarty's do a few.
The Porter house (which brews it's own beer) usually has a fairly decent stock is on Parliament Street (end of temple bar.)
Bow's bar at the opposite end of Temple bar (Fleet Street I think) used to have a great stock.
Palace bar as already mentioned has a lovely whiskey bar room up stairs (not sure on it's opening hours though).
However the Vat House (behind the central Bank / Part of Blooms hotel) is also well worth a trip just to look at some really expensive whiskey. And if you are lucky they might have a few older bottles where the prices have remained the same but are actually tempting now, but that is just a guess on my behalf.
Where to go after that I think maybe a local Spar and get a few bottles of water