Indeed it will. Though you might need to look up how to get it working with whatever OS it has (as the game dates back to the windows XP era)
In fact, the vast majority of computers being sold second hand or given away these days should be able to run FFXI. Most charities wouldn't even accept computers that are too old to run Final Fantasy XI, especially now that Microsoft has stopped providing security updates for Windows XP.
Even old DOS era computers are getting harder to sell to people that aren't hardware collectors, as DOSBOX can run the vast majority of DOS games.
Short version, as long as it can connect to the internet, it should be fine. Though be careful, as I have read that some of these school issued computers have monitering software on them, which could put your account at risk. (also ignoring any usage restrictions that your school may have as part of the agreement for giving you the laptop)