The MIDP API defines UDP datagram connections, TCP/IP socket connections
and SSL connections, that may be implemented by devices
apart from HTTP / HTTPS (but the later two may be WAP
based for example instead of IP-based).
So, yes, a mobile phone can be IP-aware.
But in terms of design, I think
you should consider
the phone app to query the web service regularly
instead of trying to send a notification the other way round.
You see, apart from considering that the IP is dynamic,
you must account that the phone's connection is not permanent.
So it looks like a more workable design to have the phone contact
your server when it is online (and maybe poll it regularly)
rather than the server checking if the phone is online and what it is its IP.
Regards,
Eduardo