Boppard to Burg Eltz

I'd give it the better part of a day. Get a VRM Minigroup ticket. Take the train to Koblenz and change to the train to Hatzenport. There is a bus (No 330, the Burgenbus) that will take you to the castle parking lot. Buses at 9:16, 11:16, 15:16, 17:16. The Minigroup ticket is 21.80 euro and covers up to 5 people traveling together for the whole day, after 9 am weekdays, all day weekends and holidays. The train journey is just under one hour.
If it is a nice day, you can hike from Moselkern station (next stop beyond Hatzenport) The hike up is an hour or so depending on what kind of shape you are in.
You can also do a combination of the 2. Trains run each direction once an hour.

The buses to Burg Eltz only run on weekends and holiday. The rest of the time you will have to hike from Moselkern.

The trip from Boppard to Moselkern will take about 1 hr 50 min. Look up the schedules on the German Rail Query webpage. Be sure to use European date (d/m/yr) format and 24 hr time.

An additional restriction on Bus 330 to Burg Eltz - service is seasonal and begins May 1.
"Burgenbus" flyer

The walk from Moselkern

jk,

If you're not able to connect with the Bus that Lee mentioned, you could also travel by Taxi (not cheap but it is an option). In addition to the Taxi firms listed on the Burg Eltz website, you could also try Taxi Ewald in Löf (have them meet you at the station). The drivers may not speak much (if any) English, so it would be a good idea to have your instructions written in German on a piece of paper, to show the driver.