My brother works for a IT company in Michigan as a systems administrator. They allow overtime but have begun a process where they approve the overtime after it has been worked every week/pay period.

My understanding is, since they are a company that offers service to customers outside the state and even outside the U.S., that federal overtime laws are the ones that apply here.

My question is, can an employee in this situation work 50 or more hours per week and legally not get paid for anything above 40?

I understand these laws don't always seem "fair" but if an employee isn't going to get paid for overtime they already worked, shouldn't the employer setup a policy where employees must be approved for overtime before they work it?

Thanks in advance for any advice!

Yes, they would be subject to the FLSA. It's not the title that is important, but the duties. The Computer Professional exemption under the FLSA is one of the most difficult to understand.
http://www.dol.gov/esa/whd/regs/compliance/fairpay/fs17e_computer.pdf

Here's a DOL opinion letter that might be on point.
http://www.dol.gov/esa/whd/opinion/FLSA/2006/2006_10_26_42_FLSA.pdf