Hi TG and welcome to our little sod.
Your EA permit will not cover you over here. You will need to purchase a rod licence issued by the Fisheries Conservancy Board for NI. You will get one in any decent tackle shop. They are available for both coarse and game fishing, so make sure you get the game one. That is simply your entitlement to carry a rod, not permission to fish.
There are some excellent day ticket fisheries near you. I have recently fished Murloc near Aghalee and it is an excellent fishery.
As well as private trout fisheries we also have a public angling estate which is managed by DCAL. Hillsborough Lake would be one of it's waters. You can buy permits for these waters at the same shop you get your licence. There are different options, either buy a permit that covers all DCAL waters, or buy one that covers a number of local waters. Again there is a coarse fishing option so be sure to buy the game permit.
My prefered option is to fish mostly the private fisheries but keep a local DCAL permit for the waters close to my home.
Unfortunately unlike England, where your EA licence runs from April to April, both our licence and permit systems run from January to January. Means that you will really only get 3 months fishing for your years fee. You need a rod licence rgardless, but could perhaps concentrate on the private waters in the mean time and buy a DCAL permit in the new year.
You are right of course, it is a strange and over complicated system which is certainly off putting to visitors, but we live in hope that it will soon be reformed.