You can change the refresh-time from cgi.conf
This only changes your screen refresh interval. It does not change the interval that Nagios is checking the plugins. So system itself makes the checks as it used to do before. If you want to get faster response to monitored object, you have to change the object configurations:
The www-site you are watching is just graphical view to nagios. The monitoring system itself runs in background and it can be running even without GUI if needed.