Can the Mirror-Mad Phantasm rescue itself?
Unfortunately that's quite against the law!  Even the laws of the spirit world...

The way the timings work in Magic are basically as follows: after you announce the beginning of combat, both players have a chance to play spells and effects.  After attackers are declared, both players have a chance to play spells and effects; after blockers are declared, both players have a chance to play spells and effects.  Once all of that is finished with, we go directly to combat damage.  There is no window of opportunity for the Phantasm to activate its ability before it deals and receives combat damage and goes to the graveyard.

Well, I say that... there is one way!  If you can manage to give the Mirror-Mad Phantasm first strike, it will deal its 5 damage in the first strike combat damage phase, and then there will be a chance for you to activate its ability before the normal combat damage step.  It's probably not what you wanted to hear, but that's the only way to do it.

Are you getting as excited for Innistrad as I am?  I can't remember the last time a Magic set had so much sheer flavour about it...