I went to ASU for graduate school. You are very wise to look into ASU for the undergraduate years. ASU has an exellent honors college. Admission is competitive, but if you are one of the best students in your HS and an Arizona resident, then you should be able to get in. I was at ASU for a long time and am aware of the high standards. One of the students in the Honors College was in my class. It's a lot of work. But you have the "home court advantage" and very low costs. Private tuition or out-of-state tuition is ludicrously expensive at the undergraduate level and not worth the price. I went to a private university as an undergraduate. I wish I had gone in-state instead.
The academic standards of the Barrett Honors College are fine. Also apply to U of A and NAU. Just be sure you choose a major that is in high demand. Some college students choose the wrong major and wind up living at home with their parents after they graduate. Strong undergraduate majors in recent years of the recession include healthcare (i.e. allied health), accounting, health economics, nursing, radiology, biomedical engineering, computer science, medical informatics, MIS, systems engineering, web design, physical therapy, etc. Do a job search on http://www.dice.com
and see what jobs interest you, then choose the appropriate major. Also try other job search sites like monster.com.
So ASU has a good honors college, but just be sure you choose the right major. If you want to attend a private university or out-of-state university, then wait until graduate school because at that point you get much more financial aid. But ASU and U of A both have strong graduate or professional schools. The Mayo Clinic also has a Scottsdale campus.