Short Answer: the warp core is shielded, but not all the time.
Long Answer: as Richard and I have discussed in the comments, the warp core is capable of being enclosed in a containment force field. While ostensibly these fields are primarily to protect the crew in the event of a breach so they can evacuate Engineering and eject the core, I suggest that containment fields are also able to absorb the energy of a hand phaser.
In TNG: The Game the bridge crew attempt to catch Wesley by blocking off corridors with "security fields". Wesley then sets up a phaser to fire at the field as a misdirection. Despite security fields being meant to control people, including perhaps armed intruders, there is no reason to believe that they are significantly different from a containment field in form or function. And as a containment field should in theory be able to absorb the energy of a warp core breach, even if for just a second, there is no reason to believe it couldn't also absorb the lower energy of a phaser.
As for why the containment fields wouldn't always be active, or automatically activate on a red alert, there is no information to my knowledge that addresses this apparent oversight in the Starfleet SOP.