C++ program code:

//header section
#include <iostream>
#include <cmath>
using namespace std;

//Create a class name, PrimeNumber
class PrimeNumber 
{
private:
    //Declare variable
    int prime;

public:
   //Default constructor
    PrimeNumber() 
    {
        prime = 1;
    }
    //Constructor with parameters
    PrimeNumber(int num) 
    {
        if (isPrime(num))
            prime = num;
        else
            prime = getNextPrime(num);
    }
   //The getPrime function returns the stored prime number.
    int getPrime() const
    {
        return prime;
    }
   //The prefix increment operator (++) is overloaded to 
   //find the next largest prime number and update the prime variable accordingly.
    PrimeNumber& operator++() 
    {
        prime = getNextPrime(prime + 1);
        return *this;
    }

    //The postfix increment operator (++) is overloaded to create a copy of the current object, 
    //find the next largest prime number, update the prime variable, and return the copy of the original object.
    PrimeNumber operator++(int)
    {
        PrimeNumber temp(*this);
        ++(*this);
        return temp;
    }
   //The prefix decrement operator (--) is overloaded to find 
   //the next smallest prime number and update the prime variable accordingly.
    PrimeNumber& operator--() {
        prime = getPrevPrime(prime - 1);
        return *this;
    } 
    //The postfix decrement operator (--) is overloaded to create a copy of the current object,
    //find the next smallest prime number, update the prime variable, and return the copy of the original object.
    PrimeNumber operator--(int) {
        PrimeNumber temp(*this);
        --(*this);
        return temp;
    }

private:
    //isPrime function checks whether a number is prime or not.
    bool isPrime(int num) const 
    {
        if (num <= 1)
            return false;

        for (int i = 2; i * i <= num; ++i) {
            if (num % i == 0)
                return false;
        }

        return true;
    }
    //The getNextPrime function finds the next prime number greater than the given number.
    int getNextPrime(int num) const {
        while (!isPrime(num))
            ++num;
        return num;
    }
    //The getPrevPrime function finds the next prime number smaller than the given number.
    int getPrevPrime(int num) const {
        while (!isPrime(num))
            --num;
        return num;
    }
};
//main method
int main() 
{
    //a PrimeNumber object p1 is created using the default constructor, and its prime number is output.
    PrimeNumber p1;
    cout << "Default Prime Number: " << p1.getPrime() << endl;
    //PrimeNumber object p2 is created with an initial prime number of 13, and its prime number is output.
    PrimeNumber p2(13);
    cout << "Initial Prime Number: " << p2.getPrime() << endl;
    //The prefix and postfix increment operators (++) are used on p2 to
    //find the next largest prime number, and the results are output.
    PrimeNumber p3 = ++p2;
    cout << "Next Prime Number (prefix ++): " << p3.getPrime() << endl;
    //The prefix and postfix decrement operators (--) are used 
    //on p2 to find the next smallest prime number, and the results are output.
    PrimeNumber p4 = p2++;
    cout << "Next Prime Number (postfix ++): " << p4.getPrime() << endl;

    PrimeNumber p5 = --p2;
    cout << "Previous Prime Number (prefix --): " << p5.getPrime() << endl;

    PrimeNumber p6 = p2--;
    cout << "Previous Prime Number (postfix --): " << p6.getPrime() << endl;

    return 0;
}

OUTPUT OF THE PROGRAM CODE:

Default Prime Number: 1
Initial Prime Number: 13
Next Prime Number (prefix ++): 17
Next Prime Number (postfix ++): 17
Previous Prime Number (prefix --): 17
Previous Prime Number (postfix --): 17