public static int getRandom(int[] array) {
    int rnd = new Random().nextInt(array.length);
    return array[rnd];
}

Use the Random.nextInt(int) method:

final String[] proper_noun = {"Fred", "Jane", "Richard Nixon", "Miss America"};
Random random = new Random();
int index = random.nextInt(proper_noun.length);
System.out.println(proper_noun[index]);

This code is not completely safe: one time out of four it'll choose Richard Nixon.

To quote a documentation Random.nextInt(int):

Returns a pseudorandom, uniformly distributed int value between 0 (inclusive) and the specified value (exclusive)

In your case passing an array length to the nextInt will do the trick - you'll get the random array index in the range [0; your_array.length)

Simple! Method nextInt(facts.length) returns a random integer between 0 and the length of array. Put it as the index of the array to get the random one:

Random rnd = new Random()
String randomFact = facts[rnd.nextInt(facts.length)];

anyItem is a method and the System.out.println call is after your return statement so that won't compile anyway since it is unreachable.

Might want to re-write it like:

import java.util.ArrayList;
import java.util.Random;

public class Catalogue
{
    private Random randomGenerator;
    private ArrayList<Item> catalogue;

    public Catalogue()
    { 
        catalogue = new ArrayList<Item>();
        randomGenerator = new Random();
    }

    public Item anyItem()
    {
        int index = randomGenerator.nextInt(catalogue.size());
        Item item = catalogue.get(index);
        System.out.println("Managers choice this week" + item + "our recommendation to you");
        return item;
    }
}

Just store the index generated in a variable, and then access the array using this varaible:

int idx = new Random().nextInt(fruits.length);
String random = (fruits[idx]);

P.S. I usually don't like generating new Random object per randoization - I prefer using a single Random in the program - and re-use it. It allows me to easily reproduce a problematic sequence if I later find any bug in the program.

According to this approach, I will have some variable Random r somewhere, and I will just use:

int idx = r.nextInt(fruits.length)

However, your approach is OK as well, but you might have hard time reproducing a specific sequence if you need to later on.