Firstly, the UK and the Republic of Ireland operate an 'open border' policy so, as far as the 'authorities' (on either side of the Irish Sea) are concerned, you don't need a passport to visit the Irish Republic (or to return to the UK).
Secondly, the '6 month rule' is largely a myth. There are very few countries which require you to have 6 months left on your passport for it to be accepted. Within the EU, all countries are obliged to allow any EU citizen to enter (with proof of ID, such as a passport or national identity card) right up until the nominal expiry time of their ID document. (i.e. you can arrive in Spain, Italy, Greece, Ireland, or any other of the 27 EU member countries, on the very last day of your passport's validity. The immigration authorities are obliged to allow you to enter).
Irrespective of immigration rules, all airlines require passengers to provide photo ID. Every airline accepts a valid (i.e. 'in date') passport for this purpose. For travel within the UK, and from the UK to the Republic of Ireland, other forms of ID may be acceptable. Exactly what you can use varies between carriers but even the strictest airlines (e.g. Ryanair) accept a photo-style EU driving licence.
To summarise:
1. You don't need a passport (or any other form of ID) for Irish immigration purposes.
2. You can use your passport as ID for the flight.
3. If you can't find your passport, you can use your driving licence instead.
Chris