The choice is yours. However, if you don't upgrade it, you need to make sure you have all the software it will ever use installed from the beginning, as installing new software on an out-of-date system is highly likely to cause problems. It also means the server will be increasingly vulnerable to security holes as these are discovered and not plugged, so you will need to ensure it is protected from the Internet by an external firewall. If you later want to access it remotely it will not be suitable for that because the vulnerabilities will make it an easy target.
The alternative is to ssh into it once a week and run apt-get update && apt-get dist-upgrade or even to set these up as a cron job. However, that carries the small risk of an upgrade breaking something or overwriting a configuration file, so you should always keep a fairly recent (say up to a month old) remaster from which to restore it if necessary. You probably can't do the restore headless.