There is not a hotel shuttle. However, an Airport Shuttle, rideshares - ex. Uber & Lyfts, etc. can be ordered by you or specific associates of the hotel. (Jay, the Doorman.)
About the shuttle.... Yes I saw a shuttle parked near the entrance with the hotel logo on it. I didn't use it, we walked most everywhere and we had a car..
Get an answer from Omni Austin Hotel Downtown staff and past guests.