Since you have already done this, you should be familiar with the process of the upgrade/migrate data from Joomla 1.5 to Joomla 2.5.
However much of the success of your attempt is depending also on the overall db changes that might have occurred on your current 1.5 live site.
Given that, you might be just good by only migrating the menu-items, but if any other changes occurred in your website, like changing older articles or categories etc (changes that do not exist on the already upgraded 2.5 site), then you might end with a messy database, if you only try to import the menu items.
In this case, you might want to consider to migrate/upgrade the whole database again, and connect your 2.5 website with this database.
But if you are sure that you only need the menu-items, then follow below:
Probably you used one of the available tools JUpgrade or SP Upgrade. I don't know about JUpgrade, but I would recommend to use the SP Upgrade (paid download), which allows migration of selected db tables.
If this is not the option for you, then perform the full database upgrade with your preferred method (as you have already done), in order to create a fresh new 2.5 database (not do this over the current 2.5). Then you can export/import/replace data from the fresh 2.5 to the old one.
Hope this helps a bit...