If you like to hike you can use one of the lower car parks and hike up through the forest paths (which is what we did without realizing there were other options) or you can park much closer to the castle and walk down a paved road. There is also a shuttle you can take for 1.50 Euro each. I would recommend walking down, and then taking the shuttle on the way back up to your car as it's a pretty steep hill going back up.
The car park we used was called Ringelsteiner Muhle in Moselkern. There was a restaurant and I believe a hotel near the car park area. The sign said 40 min but it took closer to an hour and a half with a five year old dragging behind.
If you do chose to park down below and do the hike, make sure you still go up the big hill. There is a great place to stop and take pictures. It's worth the walk.
For the official Burg Eltz website click here.
Here are some other interesting views we saw on the drive.