The "castle bus" runs to Eltz from a couple of different Mosel towns on weekends and holidays, but on weekdays there's no transport to the castle. On any day of the week you can take the train from Koblenz to Moselkern and walk through the woods from there (a very pleasant walk.) See this description of the walk, which many consider part of the fun.
"Also - is it worth it to have a car in that area in order to explore? As it happens, our reserved accommodations are not close to a train stop so that's why we're leaning toward a car. "
Trains cover the Rhine, Mosel, Lahn and other river valleys in the area, where most of the great scenery and the interesting towns and sights are. Koblenz, Oberwesel, Boppard, St. Goar, Braubach, Cochem, Linz, Remagen, Limburg, and many other worthwhile towns all lie within the same local transit zone - the VRM - and day passes for using the trains here make exploring pretty easy and inexpensive. These passes cover local buses and some ferry crossings (like the one in St. Goar) as well. Trains run hourly or better and hit all the towns on both sides of the Rhine - and most towns on the Mosel as well. A day pass for 2-5 persons costs €23 and covers the light green VRM area on this map:
http://www.vrminfo.de/fileadmin/user_upload/VRM-Schienennetzplan_2018.pdf
If you will be doing a river cruise, using the train in conjunction with that is usually a necessity anyway.
I have used a car in the area when I had my elderly in-laws and two others along, but all my other visits have been by train. I normally book somewhere near a station so that a bus ride to the station is unnecessary. If you are locked into your Koblenz accommodations, you might need the car, no way to tell without knowing your location. If you don't mind changing accommodations, there are normally plenty of nice places to stay near the railway locations. I don't recommend Koblenz, as a rule, which does have a good location for outings but was mostly flattened in WW II and rebuilt sort of hastily and is not especially scenic or atmospheric, at least not in the same way Boppard, or Braubach, or Cochem, or Bacharach might be. (Be especially careful in Bacharach, where some hotels like the Kranenturm are just way too close to the tracks.)
Braubach btw is the home of Marksburg Castle - which has regularly-scheduled tours in English and a more authentically medieval feel than Burg Eltz. Braubach lies roughly across the river from Boppard and about 10 train minutes south of Koblenz on the east bank of the Rhine.