Alicante and Elche are on the same train line. Trains (renfe) run frequent whole day long. The ride to Elche will cost € 2,15 per person one-way and will take 30 minutes only. You can find more info about the train schedules here.
How do I get to the train station from the cruise port in Alicante?
Alicante train station is located at the Avenida de la Estacion, slightly uphill, in the western part of town. It’s possible to walk there, but at taxi is the better option.
When arriving by train in Elche, you will find most highlights on walking distance from the train station.