There are a number of wi-fi spots in and around Sunny Beach, it's just a matter of running a search to access them.  In most instances the bars and restaurants will provide you with their user-names and passwords, if you ask. 

Many people though like to stick to the ones they know are safe, like Condor Bistro/Cocktail bar just along from Mania Disco in the centre of the resort.  You'll also find that Coffee Republic in the Royal Barcelo Mall provides free wi-fi access, as well as access to pc's if you've not got a smart phone or pc with you.