When a network interface is started scripts in
/etc/network/if-up.d are ran. When the scripts are executed lots of environment variables are provided that include the IP address and other network parameters. It should be very easy to write yourself a if-up script that sends out an email with the information you want.
I think that doing a dynamic dns update would probably be more useful then an email, but perhaps you have some reasons you haven't mentioned.
Here is a old script I had that did something close to what you want.
# interface definitions
MSG_OUT_UP='outside interface is up. Address is:'
[ ! -x /etc/default/if_notify ] || exit 0
if [ "$IFACE" = "$IF_OUT" -a "$MODE" = "start" ] ; then
if [ "$IF_IN" -a "$MSG_OUT_UP" -a "$EMAIL_OUT" ] ; then
if [ "$IFACE" -a "$MSG" -a "$ADDR" ] ; then
echo -e "From:root@$NAME\nTo:$ADDR\nSubject:[IFNOTIFY] Interface $IFACE up on $NAME\n$MSG" \
| sendmail -f root "$ADDR"