How can I fix the Permission error when I call session_start()?

You can modify another users session (see below), although personally, I would recommend against it. As I imagine it can open up a whole world of session hijacking and other vulnerabilities.

With your example use case

A common user is logged, while in the same time an administrator uses the Admin functions and change some value for this user. If the value is not something obtained from the database every time, the session variable for that current logged in user need to have its value changed.

You would be better of updating the value in the database and then just checking to see if it's changed before you process the next page. If you don't want to be checking multiple user fields before each page load then when you update the user in the admin panel, you can build a hash of the values and add it to a new column called session_hash. Then just compare this field on page load

But if you still want to modify another user's session, you can set your current session_id to the targets.

// End my current session and save its id
session_start();
$my_session_id = session_id();
session_write_close();

// Modify our target session 
session_id($target_id);
session_start();
$_SESSION['is_logged_in'] = false;
session_write_close();

// Start our old session again
session_id($my_session_id);
session_start();

EDIT

Example: https://www.samdjames.uk/session_example/index.php

Example Src: https://gist.github.com/SamJUK/c220e3742487567c6262238edf85695e

Check this reference. You can use jdbcTemplate.update as:

EDIT Added imports as asked

import java.sql.Connection;
import java.sql.PreparedStatement;
import java.sql.SQLException;

import org.springframework.jdbc.core.JdbcTemplate;
import org.springframework.jdbc.core.PreparedStatementCreator;
import org.springframework.jdbc.support.GeneratedKeyHolder;
import org.springframework.jdbc.support.KeyHolder;

following is the code usage:

final String INSERT_SQL = "insert into my_test (name) values(?)";
final String name = "Rob";
KeyHolder keyHolder = new GeneratedKeyHolder();
jdbcTemplate.update(
    new PreparedStatementCreator() {
        public PreparedStatement createPreparedStatement(Connection connection) throws SQLException {
            PreparedStatement ps =
                connection.prepareStatement(INSERT_SQL, new String[] {"id"});
            ps.setString(1, name);
            return ps;
        }
    },
    keyHolder);
// keyHolder.getKey() now contains the generated key