How can I handle exceptions thrown by the script in a complex apex script?

In the context of Salesforce, you can handle and even troubleshoot exceptions thrown by an apex script in Salesforce by using the several steps which are given below:-

Use try-catch blocks

You can try to wrap the code that may throw exceptions within try-catch blocks for handling the exceptions gracefully:-

Try {
    // Code that may throw exceptions
    // For example, DML operations, SOQL queries, etc.
} catch (Exception e) {
    // Handle the exception
    System.debug(‘An exception occurred: ‘ + e.getMessage());
    // Optionally, log the exception or take appropriate action
}
Catch specific exceptions
You can catch specific types of exceptions for handling them differently based on their cause:-
Try {
    // Code that may throw exceptions
} catch (DmlException e) {
    // Handle DML exceptions
    // For example, log errors and rollback transactions
} catch (QueryException e) {
    // Handle SOQL query exceptions
    // For example, log errors and notify administrators
} catch (Exception e) {
    // Catch all other exceptions
    // Handle them generically or rethrow for higher-level handling
}
Logging and monitoring
You can use system debug logs and even the built-in monitoring mechanism of Salesforce for tracking the exceptions and even identification of root causes.  

  // Enable debug logging for specific users or contextsSystem.debug(LoggingLevel.ERROR, ‘Debug log message’);

Unit testing

You can write comprehensive unit tests for your particular Apex code to ensure that it behaves as expected and handles exceptions correctly in various scenarios:-

@isTest

Private class MyApexClassTest {
    @isTest
    Static void testMyMethod() {
        // Test case to cover positive scenario
        // Ensure the method behaves as expected
        // Test case to cover exception scenario
        // Ensure the method handles exceptions gracefully
    }
}