Hardening the tip is worth doing because any wear will cause inaccurate measurements but I suggest there's no advantage in tempering as well. Hardening leaves metal brittle. Tempering has the effect of improving toughness by reducing hardness, which is important if the metal is going to be stressed. As a micrometer tip is lightly loaded and needs to be as hard as possible to reduce wear, there's no gain from tempering. Unlike a silver-steel tap or drill which is worth tempering because it needs to be both tough and hard.
Note how difficult a domed end micrometer makes it to accurately measure anything other than a tube. The screw-end and anvil of an ordinary micrometer should be the exact opposite – flat, parallel and sharply square to reduce rocking errors. An interesting question is how flat can a surface be faced by lathe? Imperfectly! As lathes that face convex are a right pain because rods made on them won't stack, most lathes are set to face slightly concave. For this and other reasons micrometers aren't made on lathes at all – instead they are precision ground.
The slight imperfections of a home-made tip may not matter unless the mended micrometer has to be really trustworthy. If accuracy matters buy a new micrometer!
Dave