You can have any weather in Ireland in January. It can be warm or cold, raining or sunny or (less often) snowy.
Here’s what I suggest.
- Long-sleeved cotton shirt
- Lightweight waterproof and windproof coat/jacket
This combination (or subset thereof) will be adequate for the full range of conditions you are likely to encounter in January. It also means that if you need to look smart indoors, you just take off your fleece and you are wearing a nice shirt.
Below the waist, you just need ordinary trousers. No long johns or thermals or anything like that.
Bring sturdy footwear that will keep your feet dry even if there are puddles or (less likely) slush.