What is the best way to get from dublin airport to cork?

The quickest way would be to get a taxi.

The cheapest (€2.20 adult single at time of writing have change, they won't accept notes) would be to get one of the Dublin Bus routes to the city centre and then another bus, possibly the 46A from the city centre. Another option would be to get a bus to the city centre and then the train, known as the DART to Dún Laoghaire. That would be quicker.

Aircoach (a privately operated bus company) now has a route that goes through Dún Laoghaire as well which is dearer(€8 adult single €14 return) but it is faster, the stop is right outside the airport door and it runs through the night. So during night time this is the cheapest option.

See the links below.