How do you think this map influenced established immigrants who saw it in the 1860s

All of the names on the map are Irish, so the map suggests that Irish immigrants may take over the United States by the 1900s.
You may well be right about people's fears that immigrants, lots of them from Ireland, would overrun the country. Look up when the potato famine was, though. The map may be from the early 1900s, but, if so, not about the potato famine in Ireland.

I hope this helps.