Backup your entire webroot folder and the database. It is eminent to save the file localconf.php in typo3conf.
You can save the database by using phpMyAdmin or using this command in console
mysqldump -a --add-drop-table -u'database user name' -p'password' -h'host' databasename > databasename_date.sql
Download here https://typo3.org/download/ the last stable version. If you have a web case with separated core-files, you only need the "Source" package. Otherwise you need the "Source + Dummy" version. Unzip it on your workstation or use "wget" on Unix based systems on the web server and decompress it there.
If you have a separated instance, copy the source directory to a place outside webroot, delete the old symlink to source and create new one.
unlink typo3_src
ln -s ../typo3_src-4.3.3 typo3_src
If you haven't separated source and dummy part, copy all the files to the webroot (overwrite the old ones).
Copy the file "localconf.php" from your backup to the directory typo3conf (overwriting).
Call the back end of your installation on www.yourDomain.com/typo3/ and log in as Admin.
Create an install unlock file (named ENABLE_INSTALL_TOOL) in the preferences of the admin user. Log in to the Install-Tool.
Click on "Database Analyser" and then the function "COMPARE".
If you don't see any obstacles straightaway, select the check boxes and press the button "Write to database". Repeat this procedure as often as it is recommended.
Go back to the "DB Check" on the left module area and check the Reference Index. Run first "Check now". If some changes could made, then "Update now".
At the end you should see what was updates and that the "Index Integrity is perfect."
Delete the entire cache and the cache files under typo3conf. These start with "temp_CACHED_ ...".