You can't, at least not with the same feature-set as the Android version.
I don't think there's any access at all to a filesystem, so you throw that File manage out (well, I guess you could keep it for uploads to Airdroid specifically and downloads from Airdroid as internal storage access is not allowed).
I don't think existing Ringtone access is allowed since ringtones are probably DRM'd up. Similarly, music is probably not likely to be accessible by Airdroid (you could probably have a remote control type thing, but you wouldn't be able to play on the remote device)
I don't think the call log is accessible either, so throw that out the window.
App list is likely also not permitted.
(not 100% sure about this, but from what I hear) You also need to
confirm access to your (for example) contact list, photos and possibly
text messages every time (assuming you're able to access those features;
I hear most of them were only introduced several months ago). Someone
feel free to confirm or deny this.
So in short: if they decided to make a port, there almost wouldn't be a point -- not necessarily because they don't want to, but because of the policies enforced on that platform. To wit, I suspect a Blackberry port would be feasible, and possibly a WindowsPhone version as well.