IIRC I did read something that said they'd done a study of voting patterns after conflicts and, generally speaking, leaders who took their country into a war did get a boost in popularity provided the war was
a) Relatively short term
b) Successful on some level
Given the UN mandate specifically doesn't allow for ground forces to be deployed, this conflict should in theory be relatively short and it should be relatively successful (the enemy has probably already lost the ability to effectively fight back from this point, and theres nothing like a war where the enemy can't fight back for turning out well) AND will be relatively low impact i.e. we're unlikely to see a lot of coffins being ceremonially paraded out of aircraft. It's pretty much the easiest bet any war could be. I doubt it's going to make him popular - people will still see they have less money in their pockets after all - but it might stave off calamitous unpopularity for a few months at least.
Jon