1

I guess you mean road running, but in case you mean trail running and can navigate at least a bit, then you could look at adventure races on http://www.sleepmonsters.com/calendar.php You'll have to use Calendar (date) and Race Finder (location) on left side of page to filter in / out appropriate races.

2

Hello tochigiken,

Where are you planning to go during your stay? There are quite a lot of races going on in April, from the South to the North. Most of the races will be near the weekend but you also can find some in the middle of the week. At the moment there are around 10 races over the country that month. Most of them have no definite dates yet but I am sure if you keep an eye on their websites, you will be able to find dates in the coming months.
Here are some of the races I found for the moment:

The Connemara Half, Full & Ultra Marathon. On the 7th April
Jacinta Obrien 10km, Limerick
Ballyheigue 10km and 5km, Kerry
Dublin City Council RR, 2mile and 4 mile
Great Ireland Run -10km
Phoenix Park, Dublin, This is Ireland's premier 10km run, and attracts elite runners from around the world. Australian Craig Mottram, won this race for its first four years and still holds the course record of 28:55
10th Ronhill Connemarathon, Galway- Half Marathon
Coole 10km Run/Walk/Jog- Galway
Butlerstown 10km and 5km, Waterford
The Good Friday Run, 10km- 8pm- Killarney
Headless Horseman Adventure Race *Ultra, Limerick- total route 45km
Ballybunion Half Marathon, Kerry
Croagh Patrick, County Mayo

I hope that this information will help you to find races where you are planning to go. ;)