These sites will give you distances and times.
The AA suggest it is about 67 miles and will take you 1h17 minutes. I'd leave 2 hours, just to be sure. And add another 2 hours to check in, and another hour to leave the car back (Dublin Airport is chaotic!!). So, I'd suggest you leave at 10.00 or thereabouts.
If you have extra time, great. But if you miss the flight....
Tullamore to Dublin Airport is 65 miles which is about 1.5 hrs drive time. If you are going to the US and dropping off a hire car, you should be at the airport 3 hours before departure allowing time for drop off plus US immigration clearance - in addition to the usual check-in and security checks.
If you are travelling to the UK or to Europe, you could reduce the 3 hour timescale to 2 hours, as you will not have to clear any customs or similar formalities.
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