The statement on the website is inaccurate.
There’s no point going to Burgh Quay. The earliest appointment you would get would probably be a few weeks away. And as the questioner says Burgh Quay only handle long-stays.
The actual position is set out in section 4(5) of the Immigration Act 2004, which provides that non-nationals entering the Republic of Ireland by land can remain in the state for up to a month without getting a permission to land, provided they:
As far as I know the only place for tourists to get an entry stamp is at an airport.
On the plus side:
This Irish Government webpage says:
What to do if you arrive via the border with Northern Ireland
If you arrive via the border with Northern Ireland, you must report to an Immigration Officer at Burgh Quay, Registration Office, Dublin or your local Immigration Office at a Garda Síochána (Irish Police) station as soon as possible.
Thus: go to the Burgh Quay Registration Office when you arrive in Dublin.
If the Burgh Quay office is closed or otherwise unavailable — a web search (see this Irish government page and this secondary source) suggests that the office is open only by appointment — then find a Garda office and go there. The Irish government has an online Garda office finder.
Credit:stackoverflow.com‘