Ok, so this isn't in Northern Ireland, but it's only a short ferry crossing and a shorter drive away. Therefore it's very easy for anyone living in NI to visit, which I did during a trip to Scotland in July 1999.
Logan Botanic Garden is located near Port Logan in Wigtownshire, about 14 miles south of the ferry port of Stranraer. The most up to date information I have at the moment is that it is open from mid March to the end of October from 10am to 6pm daily. There is a nominal admission charge.
If it's only palms you're interested in you may be slightly disappointed since they are limited to various trachycarpus species (including an avenue of t.fortunei). However, the garden also boasts a variety of plants from South and Central America, Southern Africa, Australia, New Zealand and the Canary Islands. (The garden of most interest for palm fans has got to be Tresco in the Scilly Isles - I haven't visited it but hope to someday)
Logan has a very similar climate to the Irish coasts, wind rather than cold being the greatest threat to some of the plants. Anyway, here are a few pics:
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