I live around this area and prices are very high. It's not really a student area though. It's very popular because of it's proximity to the hospitals and other businesses in the area. It is by the way a great place to live.
There are some apartments that are within walking distance to your job. They aren't fancy or anything and I have no idea what they cost but I don't think they are outrageous. Call them and ask:
http://www.lemedapartments.com/
If you don't find anything in the immediate area, I would try the Far West part of town. There are tons of apartments there. From your 38th and Medical location you would drive west and enter Mopac (Loop 1) and then head north to the Far West exit (third one up from 35th). You can also avoid Mopac altogether when you figure out the area. It's no more than a five or ten minute drive if there are no backups on Mopac.

And like I said, you can avoid Mopac altogether.
Have you looked at Craigslist? I found one example for you in the Far West area:
http://austin.craigslist.org/apa/285408162.html
Hope this helps! Definitely check out Le Med apartments. There's nothing like walking to work (and I'm talking a five minute walk).
Let me know if you need more help. I know the area very well.