We found many places in the Centro where you can access the internet and make phone calls. They are called locutorios. There seemed to be one on every block. Sometimes they are also a convenience store, selling soft drinks, candy, etc. Unfortunately, they sell cigarettes too and allow their clients to smoke while using the computers. Some of the locutorios had a non-smoking section. One is particular on Florida was non-smoking. I wish I had noted the address since it was the exception. For about 30 minutes on the internet it was one peso ($.30 US) The speed was very fast on every one we went to.
The locutorios that had phones had the phones in booths away from the smoke. I believe it cost about $5 US for about 20 minutes to the US when my husband called his mother.