What are your plans for the 12 unit property?
If you plan to buy and hold, then find out everything you can. A general inspection & pest/fungal inspection is a good place to start. They may spot a stain and recommend another specialty inspection like roof or plumbing. Speaking of plumbing, if it is an old property, is it still on septic? If so, you could be looking at tens of thousands to convert to sewer in the future. Walk the property and ask the tenants if they have noticed anything or have complaints. Is the present management responsive? For instance, you may learn from a tenant that the building blows fuses easily in which case you probably want an electrician to see what's going on (old wiring not able to keep up with modern demands?)
You can also go down to the assessor's office, type in the address or apn, and view the title history (take a smart phone and just take pictures of the computer screen). Make note of the tax rate and use it to estimate what the tax bill will be after the sale.
Then go a couple blocks to the building department and peruse the building file. Take photos and/or make a list of when permitted work was done (don't be surprised if that "10 year old roof" was replaced in 1997...) Sometimes you'll find complaints & violations in this file -- look for a pattern of long-term issues and/or neglect.
Check out the flood maps and see if it is on a flood plain. Check out the city (re)development plan and see if they have any changes planned nearby. For instance, there is an old 8-unit complex on 9th & Willow where the city/caltrans plans to put in a sidewalk. This new sidewalk will eliminate all of the existing "off street" dirt parking and probably make the property less appealing for commuters.
Talk to your insurance agent and see how much insurance is going to run.
Talk to the existing property management and get a feel for them to see if they deserve your business in the future. Get all the leases and take a look at the rents and terms. See if that property management company is keeping up to date on the latest laws (for instance, look to see if a newer lease still requires the tenant to replace batteries in smoke detectors -- it's now the California Landlord's responsibility). Try to get a payment history -- do any tenants have a history of paying late? If there are any below-market rents, how long must you wait before their lease expires? Is the turnover rate reasonable? Obviously turnover will be higher if it is student housing, but if tenants rarely renew their lease find out why.
Check out the property management statements for the last few years -- is it really cash flowing? If there have been any big repairs, see what was done & get a copy of the receipts. Were permits pulled? If it hasn't been under property management, do the owners keep good records? Do you feel they are being honest and forthright?
Good luck!