Since you are using outlook Web Access, you are definitely using Exchange. And from the looks of that screenshot, I am going to assume that your school actually uses Exchange 2007, but I am probably wrong as some of the UI is looking a bit dated, which might mean you are on an Exchange 2003 server.
Activesync is the feature of Exchange which allows you to use an iphone with Exchange server. Activesync is a protocol which Microsoft has licensed to Apple. Apple is basically in the process of modifying their iphone software to be able to utilize this protocol. In this, you will be able to not just get mail, but more important - contacts, and calendar for that matter (also tasks...but who uses tasks???)
So, in the end, you will just need to ask your IT staff if they have:
1. Enabled activesync
2. If they are on Exchange 2003 or 2007 (2007 has more security policies)
Also, if they havent enabled activesync - ask them why not. Activesync is perfectly secure, and its basically a switch that just gets turned "on"
If they have allowed OWA, there would be no reason not to enable Activesync.