ILO and DNS – how does it work

You cannot use the DNS to point to a port (unless the client supports SRV records, most don't).

Websites and Protocols with Host Headers

You will have to put some front-end method in place to do this. Typically you would use a front end web server or a dedicated proxy software to forward the connection from port 80 to port !80 based on the name of the server being requested in the header. Some firewalls can also forward based on the host header too.

SRV Records

Some clients support lookups of SRV records which indicate hostname and port number of server for the specified service (ie the user specifies "example.com", the client looks up a SRV record and gets "server101.example.com" on port "255"; then connects to that). Some clients also implement this where it is not required (my last smartphone would lookup the SRV records when setting up a new e-mail account for example).

Unfortunately support for SRV records is highly uncommon. Only a few notable protocols mandate it's support (Jabber/XMPP, Kerberos, LDAP, SIP) and not every client supports it even when mandated.

When you first turn the server on, there's a a section during startup that says "Push F8 to enter iLO Configuration" (might be some other function key). Do that and fill in the network information and setup a login.

Once that's done, open a web browser at your computer and enter the IP you configured. It should be very self explanatory from there.

The iLO configuration is not stored on disk, and not in the BIOS, so there's a decent chance it isn't clear. Also, how your VPN works and how to use it is completely outside the scope of using iLO.