Burg Eltz from Boppard

If it is May to October, there is a bus from Hatzenport train station on Saturday, Sunday, German holidays.
https://burg-eltz.de/en/planning-your-trip-to-eltz-castle/arriving-by-arriving-by/arriving-by-public-transport.html

You get to Hatzenport from Boppard by train, with a change at Koblenz.
https://www.bahn.com/en/view/index.shtml

You can use a VRM day ticket. The minigruppen ticket is 23 EUR for up to 5 people traveling together. Unlimited travel on the regional trains and buses for a day.

We don’t mind hiking some, just not the whole way.

I don't know if that is a cheeky comment, or if you are open to hike there from Moselkern train station. Its about 1 1/2 hours (I'm a slow walker), steep trail part of the way in the forest. Don't try it on a rainy day. But that is how to get there on a week day.

We don’t mind hiking some, just not the whole way.

Hopefully by "the whole way", you mean not all the way from Boppard. You can take the train (change in Koblenz) to Moselkern Bahnhof and hike from there to Burg Eltz. I think the hike is something like 1¼ hr, 3 miles, . A lot of people to it that way.

The bus from Hatzenport is a "Rufbus", which is a bus that runs on a scheduled route, but only if someone wants to take it. You have to call a number in advance to reserve it. If nobody calls, it doesn't go. Go to the Bahn query webpage and put in "Hatzenport" to "Wierschem, Burg Eltz (Tourismus)" and find the number under "details" for the connection. I wouldn't count on someone at that number speaking English. Maybe someone at your accommodation can call for you. Or do the 6 mile (RT) hike.

Depending on the difficulty of arranging public transportation, possibly consider renting a car for the day. You can drive right up to it and not make the hike. Just an alternative to consider. We loved the castle.

Took the train to Moselkern and hike in. Gorgeous!! And great exercise!! 😊Thanks everyone!!