Alternative to Mary Gibbons Newgrange tour?

Assuming you can buy tickets online for the day you want to go, you don't need a car to get there from Dublin. We are in Ireland now and just visited newgrange yesterday.
There is an express train from Connolly station in Dublin that takes 40 min to get to Drogheda which is the closest train stop to the Bru na boinne visitor center.
You can take a taxi to Connolly station (once you check the train schedule), and its very easy to buy a day return ticket- it was 12 euros pp.
Once you get to Drogheda there is no taxi stand but taxi drivers have left their cards on the wall of the train station. We called Dee's taxi and he was there to pickup up in 10 min.
The Bru Na Boinne visitors center was a 10-15 minute taxi ride.
Once you get there and check in, you go through an exhibit and then take their bus system to your knowth and newgrange and back. It was amazing! Highly recommend.
In addition, you can download the app FreeNow, and it is very useful for taxis in the Dublin area.

Annemargaret -- thank you -- this is very helpful; so amazing that you were just there yesterday!!! I see that the tickets are sold online no more than 30 days ahead. The only day we can go is a Sunday, and it appears that most timeslots for weekend days are sold out pretty quickly. Scoring the tickets could be a challenge for us! So I'll have to find out when the sales window for our date opens.

Did you feel that 45 minutes was enough time to see the exhibits?

Did you try to visit any other sites in the area, such as the Hill of Tara?
Thank you again for this useful reply!

Yes, try to go online the day the tickets open 30 days ahead. I don't know where you're located but with us the time change was enough that I could do it the night before from California.
I felt that 45 minutes was plenty of time for the visitor center exhibit. After that you can also get a coffee or something at the cafe. Then there's a 5-minute walk to where the bus will pick you up to take you to knowth and then newgrange!

Yes, this is a question I have also, what about getting to the Hill of Tara without a tour? Can one catch a taxi between Hill of Tara and Newgrange?

There is an app called Free Now. You can use it to order a taxi in Ireland. So if you're somewhere where there is no taxi stand it comes in very handy.

You might try Paddywagon tours out of Dublin they go many places.