I am not recommending this GOTO approach: as others have already said, exceptions are the correct way to handle errors in PL/SQL.  But to address your specific question, you could do this:
BEGIN
  IF V_SYS_ERROR <> 0 THEN
    GOTO SQL_ERROR;
  END IF;
  GOTO PROC_END;
  <<SQL_ERROR>>
    V_SYS_ERROR_MSG := SUBSTR(SQLERRM, 1, 252);
    DBMS_OUTPUT.PUT_LINE('ERROR IN EXECUTION IN PROCEDURE');
    DBMS_OUTPUT.PUT_LINE('THE ERROR CODE IS ' || V_SYS_ERROR || '- ' ||
                         V_SYS_ERROR_MSG);
  <<PROC_END>>
  NULL;
END;
Of course, this still involves changing the code, so if you are doing that why not do it properly anyway?  i.e.
DECLARE
  SQL_ERROR EXCEPTION;
BEGIN
  IF V_SYS_ERROR <> 0 THEN
    RAISE SQL_ERROR;
  END IF;
EXCEPTION
  WHEN SQL_ERROR THEN
    V_SYS_ERROR_MSG := SUBSTR(SQLERRM, 1, 252);
    DBMS_OUTPUT.PUT_LINE('ERROR IN EXECUTION IN PROCEDURE');
    DBMS_OUTPUT.PUT_LINE('THE ERROR CODE IS ' || V_SYS_ERROR || '- ' ||
                         V_SYS_ERROR_MSG);
    RAISE;
END;
By the way, DBMS_OUTPUT.PUT_LINE is not suitable for output of error messages in a production application system.  Only use it for debugging during development.