Make sure it works properly.

Don't want anything sitting empty or or not properly secured.

I feel with proper management and minor updates, rents could be $750+.

In the end, it's important to properly value your time, and it's generally not effective for you to be doing hands on rehab work. 

The property needs about 5k in unanticipated repairs based on the inspection, there is a special assesment of $165 that was not disclosed in the contract, the roof on the building needs to be replaced and the association does not currently have the funds to repair it properly hence the special assessment.Long story short I don't want to walk away from this property because these are all issues that can be addressed so I countered with a lower price based on this new information.  

Fiberglass tubs should be installed with proper support underneath (e.g. mortar bed).

Then look for a local company to property manage it after you do your proper research and due diligence.

What steps could I take to make sure my property manager has the proper tenant screening criteria in place?

If you plan properly (i.e. expect annual turnover); stay on-top of things (at least quarterly inspections); and renovate for durability (students generally do not keep tidy homes and are consequently at the higher end of the wear-n-tear spectrum), student rentals can be very viable.That said, I would not plan on sharing a 4-plex with student tenants while you have a new baby ;-)

They were just so old and in such shabby condition that I was concerned with the maintenance costs over time not fitting into the pricing properly.