No train from EZE... and as far as I know, also no direct bus service to La Plata either... although you can take a combination of Colectivos from EZE to La Plata.... but I would not recommend this.
I suggest you check with Manuel Tienda Leon or similar services that are available in EZE.... MTL has published fares for private cars to La Plata.... I dont know the current prices but Im guessing it will be approx AR$250 or so....