I used to work as an unclassified academic staff employee at UW-Madison. I was a salaried manager and I filled a few classified staff positions while I was there, so I know about both types of positions. Academic staff positions always require a college degree and many of them require an advanced degree. They have the same benefits and most of the same perks as faculty positions.
Most of the positions that come open there are for classified staff, which are service and clerical type hourly positions that may or may not require a degree. Most of those also have compulsory union membership. The union contract places a lot of value on seniority. If you are new to the system and get a position during a time when another department is laying off people in the same position category, you can get bumped out of your position by a more senior union member. That union member can take your job and you can either bump someone else with less seniority than you or lose your job and have to be rehired by another department. The prime new opportunities usually get filled quickly through this bumping system.
Usually the work hours of full-time classified positions are a very rigid 7:45-4:30 schedule Monday-Friday. It sounds like that schedule will suit your needs. I was flexible as a manager and let my people work a mutually agreeable schedule that took them out of rush hour traffic.
As a manager, I despised the bumping system, because it replaced a couple of good employees I had carefully selected with what can best be described as dead wood types. I can imagine that new employees must live in fear of it, too.
You can look at current open positions and read more about the two types of positions
here. I would also recommend looking for open positions at UW-Extension, which has openings on the campus and around the state posted
here.