Bratislava is pretty much like most former Eastern Bloc locations in my experience.
Especially in cities, most people who were teenagers at the end of communism have learned English to some degree. Most older people have not.
This means there is no shortage of people up to about age thirty with pretty good English who can fill the jobs in the tourism industry: hotels, museums, etc. When you need to communicate with random people not in tourist-facing jobs, pick younger looking people.
In country areas outside Bratislava it will be a little more difficult.
And obviously if you know any Slavic language (Croatian, Polish, Russian, etc) you will be able to find a way to communicate with anybody who doesn't know English pretty easily. Asking for directions or help with a flat tyre doesn't require the finesse of translating poetry or being a professional interpreter after all. There's just a bunch of differences in pronunciation and vocabulary you might find to quaint, cute, funny, or annoying - and they will think the same of your speech. Keep out the slang and don't talk too fast. Polish and Slovak are relatively close even amongst Slavic languages I believe.