Special exhibits normally have timed entry. Permanent exhibits do not. Since the Art Institute's ticket encompasses both - as opposed to some museums where you have to pay extra for each special exhibit - that's probably why they are asking you for an arrival time.
Inasfar as the re-entry, I suggest that you send an e-mail to the AI and ask them if you would be able to re-enter ***on the same calendar date*** as shown on your paid ticket. I'm doubtful on that but you can try. But I'm VERY positive that they will not allow re-entry within a 24-hour period. You pay for each day, unless you're interested in paying for a membership.
However, even if they do allow same day re-entry, IMHO, that just isn't a wise use of your precious time in Chicago. Wherever you go from there, you would have to allow for travel time there *and back*; and that could run into hours, especially when you figure wait times for public transportation (Sunday schedules have less bus routes and fewer busses). The AI is only open from 10:30 a.m. to 5 p.m. (7.5 hours) anyway and you should plan on at least 4-5 hours there.