No, it's not possible to upgrade an Aer Lingus-metal itinerary with MileagePlus miles.
You can redeem MP miles for an itinerary that includes Aer Lingus flights in business class.
You can book an itinerary with the Transatlantic segments on UA and then do a miles+copay upgrade (20k+$550 each way, I think) if it's available on the UA segments. You can also upgrade on Star Alliance partners (of which Aer Lingus is NOT an example), but only on the limited fare classes listed here:
http://www.united.com/web/en-US/cont...rade/saua.aspx
Happy travels!