You'd have to do the array re-sizing outside of the class in this function:
1 2 3 4 5 6 7 8 9 10 11 12 13
int noOfCustomers = 0;
Customer cust[1];
int cID;
cust[noOfCustomers++].addNewCustomer(); // Note, we can combine lines 5-6 above
cout << "Enter Customer ID to delete > ";
cin >> cID;
for (int i = cID; i+1 < noOfCustomers; ++i)
cust[i] = cust[i+1]; // Shift everything down
cust[noOfCustomers--].deleteCustomer(); // Delete the last customer by zeroing the data
use a variable to save the first empty element in your array.
if you want to add a new element you just pass the id, add your element and increase the id.
if you want to delete something, you decrease the id. the element is not really deleted that way, but it gets overwritten the next time you add something.