Ireland in January

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Wet and cold, but probably almost tropical compared to Fairbanks! Also, NW Europe often experiences very heavy fog during the cooler months. This can make driving a bit more hazardous (and navigation more confusing), but it won't be a game-stopper.

Daylight hours will also be longer than Fairbanks, but still short enough to limit exploring in the countryside.

Emily-
I spent a few weeks in Ireland in the winter of 2000/2001. It snowed in Dublin, which put everyone in a cheerful, Christmassy mood.
Expect plenty of rain and ice, and take your time driving. You'll look at a map and see a 60 mile trip, and think, "Oh, just an hour away." Not so. Usually, it will take twice that time.
It helps to have a good co-pilot with an accurate Michelen map to guide your journey.
Pack waterproof, layered clothing (as I'm sure you're used to), and spend plenty of time in pubs, making new friends and hearing great traditional music. Have fun!
-Kyle

I haven't been there in January but I have been twice in February. In both cases, it was relatively warm (warmer than Illinois), with a few cold days. There was no snow or rain, only a bit of mist. Except on weekends you won't find much music in the pubs, unless you are in Dublin. Some attractions are closed or have reduced hours during the winter. The tradeoff, though, is no crowds anywhere. The weather in Ireland tends to be warmer than you would expect in the winter (there are palm trees in some areas, after all).

Thanks for all of the great information everyone. I can't wait to go!

It may BE warmer on the thermometer, but may FEEL colder than Alaska! It is very damp and the cold seems to climb right up your legs! Be prepared to wear warm and waterproof clothing.