If i want to become a doctor, can i go to University of Texas at Austin?

Yes. You can do your 4-year college degree pretty much anywhere. You need the degree to apply to medical school.

If your question is for undergraduate study, then yes, you can. You can attend any accredited school for undergraduate.

If your question is for medical school, then yes, you can also go to UT Austin for med school.

EDIT: Sorry I think I was wrong for UT Austin having a med school.

For undergrad, yes. UT Austin does NOT have a med school. The University of Texas System has a few med schools like Galveston, San Antonio, and Houston... Hook 'em Horns!