Internet cafes in Azerbaijan are called "internet klubs" and they are found throughout Baku. Some coffee houses provide wireless internet, but they are quite rare in Baku. Recently public wi-fi service was implemented the central areas. One "klub" with fast internet downtown is VIP, U.Bunyadzade 3. They also have a lot of games you can play.
English and Russian are spoken in many places throughout Baku, but it wise to still either bring an Azeri phrase book and/or brush up on your Azeri/Turkish/Russian before arriving. However, about 80% of population at least understands Russian, and about half the people under the age of 35 will speak at least a little English.
English is well catered for in most shops, restaurants and bars as a result of 15+ years of expats from the oil industry. Very easy to make yourself understood with little local language skills.