First of all I would like to say that following Joomla instructions for the upgrade of a Joomla 1.5 installation to Joomla 2.5.4 website is recommended.
However, I personally think it is better to make a copy of your production website to another new root folder and call the folder something like: copyoforiginalsite.
Update to the latest J1.5 version first, then make sure that jUpdater updates that version to J2.5.X
1, Either, create a zip file of your original root folder and move to to your new copyoforiginalsite folder. Or, in cPanel, select all and use the 'copy' function'.
Note: Some servers will change configuration.php file permissions to  during an upgrade or if moved, this is to protect exploits. You will need to change this back to  first before editing.
2, Update your root folder robots text file to ignor the new folder to help keep it away from search engines.
3, Download a copy of your original database, saving it locally.
3, Create a new data base for your copy website with all administer rights. Then import the database you just saved.
4, Go to your new folder (copyoforiginalsite) and unzip the copy of your website and open the configuration.php file to edit it in the cPanel Code Editor (which will make life easier because of the line numbers). This part is very easy:
var $log_path = '/###/####/public_html/logs'
var $log_path = '/###/####/public_html/copyoforiginalsite/logs'
var $tmp_path = '/###/####/public_html/tmp';
var $tmp_path = '/###/####/public_html/copyoforiginalsite/tmp';
var $ftp_root = '/public_html';
var $ftp_root = '/public_html/copyoforiginalsite';
var $db = '######_your original dadabase name';
var $db = '######_your new database name';
Save the changes.
5, Now you will have a fully functioning copy of your orginal website that you can upgrade to J2.5 following the Joomla upgrade instructions.Thinking out of the box, just think of your new folder (copyoforiginalsite) as your root folder for the upgrade
. Now if anything goes wrong during the upgrade at least your production website is uneffected. If the upgrade is successful, you can now at least make sure the new site is fully functioning and updated where neccessary before finally moving it across to the root folder.
Before the upgrade, turn off the system cache plugin and then delete all cached files.
A further note on Third Party Extensions and Templates
Make a copy of all third party extension settings
as you will no doupt need them if you need to install J2.5 version of the extensions.
The J2.5 upgrade process creates a different category structure to J1.5. Where in J1.5 you had a Section and categoried assigned to that section; in J2.5 the section becomes the top category with old categories under that. So, Your menu structure will need modifying. This will be evident as soon as you take a close look. Also, if you had 'Uncategorized' pages as child menu items, you might find these have been assigned differently. So check all this out.