This depends on why you are buying the property. I assume, since you are asking about school districts, that you are buying to live in, instead of as a rental investment property.
In Austin, the hottest neighborhoods will always be the ones close to downtown and the University. The homes are charming, the commute is shorter (if you work close-in) and the culture is fantastic: lots of interesting restaurants, the University's offerings, Town Lake and Austin's great energy. Think about neighborhoods like Barton Hills , Bouldin Creek and Zilker in 78704, Brykerwoods or Tarrytown in 78703 and the neighborhoods north of campus, like Hyde Park and even up around North Loop. The schools in these areas are some of the strongest in the city.
Austin's market is shifting but it is still very strong and vibrant.
Hope this helps you!
Kimberly White Erlinger