In my experience, you do want to exclude air and water, but you don't need to be particularly rigorous. No distilling of solvents or blowing nitrogen or argon into the reaction. The biggest issue I ran into was using old triphenylphosphine, because it oxidizes over time, so you need to add more than you think to compensate. Incidentally, getting rid of triphenylphosphine oxide is why I hate Mitsunobu (and other TPPO generating) reactions.
Order of addition is important, because you need to activate the DEAD/DIAD/whatever you use. PPh3 and azadicarboxylate first, then your substrate, nucleophile goes in last.