You’ll be dealing with 2 jurisdictions with different requirements, NI and ROI.
To bring a rifle into ROI you will need a visitors permit, no big deal you can apply on line.
To hunt deer you will need a deer hunting licence, theres a requirement for a “ national deer hunting qualification” , DSC1 or 2 or any other qualification covers it. Again, you can apply on line.
I’m not sure what the situation is for your transit through NI, you may need a visitors permit for there too, best to check, NI is officially part of the UK but has some very definite ideas of its own regarding firearms, so check.
One last thing, check the ferry companies T’s +C’s carefully, at least one company wouldn’t carry ammunition but had no problem with firearms.
Where about are you going?