We spent a night at Teresa's based on the positive reviews of others, which on re-reading describe the experience very well.
We took the (excellent) train from the airport to Bari Centrale, then walked about 12 minutes to the apartment. We'd told Teresa at what time we were landing and when we arrived and were looking for the door she leaned out of the window and called out to us. Within a few minutes we were set up with internet access, towels, a map of the city and breakfast coupons (for a coffee and cornetto at a decent café a stone's throw away), after which Teresa handed over the key and told us to ring any time if we needed her.
The apartment is on two small levels connected by a somewhat tricky staircase. Don't expect something fancy, it feels a bit dated (I couldn't get the warm water to work, but I may have missed a button) and the walls could do with a paint job; however, it is in a nice old building and has a homely feel.
The main reason to recommend the place is its excellent location, inside (though not really in the heart of) the mediaeval district of Barivecchia. My 2007 guidebook describes it as a somewhat dodgy area, but if that was true then, it isn't now; it has lots of character and is clean and well maintained, a great place to wander around and get lost. Around the corner we found old ladies making orecchiette (the local pasta) in the street, and all over families had opened up their kitchen doors onto the street and were enjoying the fine weather. We saw a few tourists but it's definitely not overrun.
There are some nice restaurants and a good gelateria close by (we had a good and cheap meal at La Tana del Polpo just down the road).
Overall we had an easy and enjoyable experience and would consider staying here again.