@Dan is correct that the access to the Natural Reserve is restricted and that you must ask for authorization prior your trip. Below are the contact details. The entrance is free of charge.
Raggruppamento Carabinieri Biodiversità – Reparto Biodiversità di Caserta
Corso Trieste 8, Caserta
Mail firstname.lastname@example.org. PEC email@example.com.
Tel. +39 082 3354693.
The following documents have the days listed when it can be visited plus some more info in Italian:
This is the official site: https://www.caimontilattari.it/sentiero/325/ (schedules for June, July, August and onward should be posted at this link).
The walk is very peaceful and in the summer months very refreshing as well. If you go there in July/August you will quickly understand why. It feels like a tropical forest. This is due to the very unique terrain configuration and conditions that created an unique ecosystem in an unusual location for such a place.
If you have any questions, feel free to ask more.