Take a yellow cab from the airport, Walk, get a ride from my AirBNB host, hail a taxi, borrow a bike, get a ride from other friends, hang out on Rainey Street with whatever friends are around and use the Getme app it get to the airport.
Lack of Uber did not affect my visit at all. If anything walking around more enhanced my experience. What a great city.
I've personally never been to Austin, but I doubt the city is going to suffer any major decline in business because Uber and Lyft have left town.
There are hundreds of cab, electric cab, and pedicab, drivers who all follow the rules. The rules may not be perfect, but they aren't particularly onerous, either. It's about $40 to get your fingerprint background check done. The rules have been in place for years for those other kinds of drivers. Uber/Lyft wanted special treatment...they wanted their competitors to keep playing by those rules, while they continued to be exempt from them.
The process they were fighting to avoid is a fingerprint and background check that costs about $40 per driver. Their millions could have paid for a couple hundred thousand drivers to get checked.