Hi Sebastian,
It is very nice to hear that you are planning a trip to Ireland this summer, and have some of the historic sites on your bucket list. You are right, Monasterboice is tucked away a little, and it can be a little detour to get there. (I couldn't find a direct option to get there by public transport either.)
However, in this article about
Monasterboice, you can find the contact details for the Drogheda Tourist Office. They organise guided tours for the monastic site and might be able to point you in the right direction for the easiest way to get there without a (rental) car.
If you plan to travel around the island by public transportation, I would recommend the
TFI Journey Planner to you. For anything that isn't accessible via public transportation, Free Now, Uber and Lyft (Dublin) are useful apps to have. You can find all links - and a few other handy app suggestions - in
this article.
Which other spots do you want to visit this summer? Do you have a little bit of time for your trip to do some exploring?
Have a nice day,
Sarah