Something in the Brushy Creek/Avery Ranch area may be doable for that budget. Mopac would get you there, or McNeil road with no toll, most mornings, in under 20 mins, if you left by 7:00AM. You may not find new construction for that cost.
I live in BC/AR area (RRISD schools and Williamson County, not in a city) and work near Mopac and Far West (just South of the Domain by a few miles). My commute most days, without bad rain and not at 7:45AM traffic, is about 20 mins. If I am travelling during non-traffic hours, I am 12 mins one way door to door. On bad days, that ~10 miles takes me an hour. Such is Austin.
If you can leave later, around 9AM, is it almost always 15 mins or so. I often work from home for an hour or so, and leave for the office later.
I choose to trade away the higher tax, higher crime, higher homeless population in Austin, for a slightly longer commute to work, and live in a safer place in a better school attendance zone. Of course, I don't really venture South of Mopac and 2222 except for maybe once a month or so, so I could care less how far away I am from downtown Austin. I am also not a bar rat, as I had my fill of random hours spent drinking at the Chicken in college. YMMV, as will everyone's, based on what they value more in life.
No matter where you pick, there will be pros and cons of both. At one point since living in the same location, my office was at 620 and Parmer. Depending on the 2 lights I had to get through, I could be home in 3 minutes. Now that was the best of both worlds, but the job there dried up!
Class of '93 - proud Dad of a '22 grad and a '26 student!