The download manager of FireFox always picks the home folder to put files in (unless you change this setting). Your home folder is called 'pbuser' which is listed the file manager's tree of items (file manager is also on the toolbar). The file manager is not as easy to use as some of the other desktops (Gnome, KDE) but it is small and that is what we need to keep the space down.
In file manager, click the right arrow beside 'pbuser' to see all downloaded files. To test the file manager; navigate to the 'code' directory and double click the file named 'pbinstall.pb' - this will open SciTE and load the file. This is the code for the PureBasic installer.
If you need more power, there is also a XFCE terminal app for ultimate system access!

My PureBasic updater app looks in the home directory by default so you should be able to just download the file, click browse on the updater, and you should see the file there.