This works the exact same way as a new charge, unless it is a probation warrant. A bail bondsman can take care of it so you don’t have to go to jail. This is called a walkthrough. If you have an active warrant that you know about, taking care of it this way is a good idea.
You will arrange with a bail bondsman or attorney, and clear your warrant without going into jail. The bonding office at Travis County Jail will book you in, fingerprint you, take your mugshot, and check for any other potential warrants. While this does count as an arrest, you only have to sit in the bonding office for a short period of time and then you are able to leave.