Dublin Airport to hotel | Taxi? Train? Bus? Should I rent?

I'd probably opt for a taxi. It is about 1/4 mile walk to the red line tram stop, so getting into central Dublin is a snap.
If you are getting a flight to U.S., you have to clear U.S. customs and immigration at Dublin airport so I'd say get there at 5 for a 7:35 flight.

Thanks Sam, Taxi does seem like the easiest option. Probably not the cheapest but that is ok. Our flight from Dublin will go back to London before we go home to the US, we will have 3 days in London prior to Dublin as well. Boston to London, London to Dublin, back to London, then home to Boston.

The train station is near the hotel, so that will be good for travel in and out of central Dublin. I'll look to see if I can go outbound anywhere interesting as well for a day trip.

With only 3 days in Dublin and you say you weren't planning on trips outside of Dublin anyway, I'd probably not rent a car simply to get yourself to and from the airport.

What does your hotel suggest for this route? On Rome2rio.com I see that each public transportation option involves 2 modes of transportation plus walking for 6 minutes. Perhaps the hotel has a different recommendation.

You ask about when to leave for your next/return 7:35 AM flight. If it is an international flight you are generally asked to be there 3 hours ahead of time. You might check with your airline for this specific airport.

If it were me, I think I'd budget for a taxi upon arrival and have the hotel book an early taxi for you on the day you leave. One tram, bus, walk option takes >30 minutes, another option takes > 1 h and a taxi ride takes 20 minutes.

Could another option be to change to a more centrally located hotel?

Unfortunately the hotel is locked in, it was part of a package I bought. The hotel website suggests this.

"Simply take the new Dublin Airport to Portlaoise Dublin Coach stopping via the Red Cow Luas stop which is only one short stop from the Kylemore Luas Stop where the Sheldon Park Hotel is located.
Fares for this service are For more information, please visit www.dublincoach.ie. "

but I think perhaps I'll just pony up for the taxi. Thank you!

The red line is just a commuter light rail operation. It doesn't go that much further out of town, but it does go to Connolly Station in Dublin where there will be more options for out of town trips.

To paraphrase Audrey Hepburn, "A taxi is always a good idea". We did that on our last big trip and it was so much easier than trying to do public transport. There is a lot to be said for getting dropped off right at the door. Have a great trip!

Thank you all!