You could install it as a server, yes. Specifically, you'll need to have packages for a webserver installed so that you can monitor and administer Nagios through the web.
Personally, I just do a "Custom" install of Fedora and install whatever packages I think I'll need for whatever role that machine will play in the company network.
So basically, how you choose to install Fedora depends on your needs and how comfortable you are with Linux.
And then, yes, you'll need to install Nagios. Best option --> from source.