Can I convert a 2 prong outlet to a 3 prong with ground at the box?

I recently moved into an older house (renting), and most of the outlets are 2 prong. I want to have a couple converted to 3 prong grounded outlets to protect my electronics. I've tested them with a light-up circuit tester, and it appears the outlet boxes are acting as ground (the tester lights up when I put one end in the small slot and the other touching the screw in the center of the cover plate). Should I be able to have an electrician just swap in 3 prong outlets and ground them on the box, or would that not meet code? I doubt I can convince the landlord to rewire the whole house, but the cost of swapping a few outlets shouldn't be too much if they don't need a separate ground. There is one outlet in the house that looks to have been converted by running a ground wire directly through the floor (and presumably a pipe or something under the house), which seems ghetto but perhaps necessary?