The only way to stay in Ireland beyond the 90 days is to apply for a Working Holiday Visa should you meet the requirements.
On 24 September 2008, the Minister for Foreign Affairs, Mr Micheál Martin T.D., signed a major new agreement with the US Government that will enable Irish and US citizens to work and travel in each other's countries for up to 12 months. The agreement reflects not only the close historical and cultural links between Ireland and the United States of America, but also the vibrancy of the modern economic and commercial relationship between our countries.
In order to qualify for the programme participants should be either in post-secondary education or have recently graduated (ie within the last 12 months).
This new programme will not affect the operation of the highly successful Student Work and Travel programme which permits students from the US and Ireland to work and travel for several months every summer. This J1 programme will continue to exist as a separate, more limited programme.
HOW TO APPLY?
US citizens
US citizens wishing to travel to Ireland under the terms of the agreement should make an application for a US Working Holiday Authorisation at the Embassy of Ireland in Washington or the Consulates General of Ireland in Boston, Chicago, New York and San Francisco.
A completed and signed application form (PDF 35kb) should be presented at the Embassy of Ireland in Washington, or at the Irish Consulates General in Boston, Chicago, New York or San Francisco together with:
• Valid United States passport;
• 2 recent identical passport photographs with your name on the reverse;
• Current curriculum vitae (with references);
• Original bank statement showing that you have access to €1,500 (or equivalent) plus a return ticket; or €3,000 (or equivalent);
• Originals of any qualifications obtained or letter from school/college/university (where applicable)
• The relevant fee of €250 or dollar equivalent as advised by the relevant Mission;
Once the Embassy or Consulate General is satisfied with the documentation above, the application will be approved in principle and the applicant notified. The Working Holiday Authorisation will be issued on submission by the applicant of:
• Return airline tickets;
• Certificate of medical/travel insurance valid for the duration of the trip (based on dates on tickets)
• Valid United States passport that is valid for the entirety of the trip to Ireland ie for a full year after their ticketed date of entry.
Source(s):
http://www.embassyofireland.org/uploads/documents/Embassy/Washington%20EM/consular%20districts%20in%20the%20us.pdf ..
http://www.dfa.ie/uploads/documents/Consular/09_01%20us%20working%20holiday%20authorisation%20application%20form.pdf ..