All of the basic skills, techniques and methodology that you learn on the CELTA, although presented in a "general English" context, are transferable to EAP and ESP contexts.
When you start to teach it's a question of adapting these skills and techniques to the different teaching contexts you're in. But remember that you'll also have to adapt to different contexts within the "general" English context. Group classes versus one-to-one classes, high levels versus low levels, French native speakers versus Japanese native speakers, students with different backgrounds, expectations, preconceptions of what learning involves... all of these present different challenges that require you to adapt your approach, use more or less of a certain technique, and so on.
So if you look at it in this way, teaching business English and other ESP contexts are just additions to the list above. They require you to adapt, throw some things out the window, do more of one thing and less of another... Sometimes teaching a business English group is very similar to teaching a general English group, but the topics and situations you use for your classes are about company ethics or marketing rather than holidays and hobbies.
It's true that some contexts are a lot more specific and require a lot more adapting of your approach than others - you may have, for example, a director of an IT company who needs English to participate in conference calls with colleagues in the UK. In this context you'll probably focus a lot more on listening comprehension, vocabulary in an IT context, perhaps language for functions such as reformulating, clarifying, interrupting, and a lot less on grammar. Your classes may start to take on more of a "coaching" role, but still, a lot of the basic teaching skills and methodology still applies.
Additional qualifications in teaching business, or young learners, or exam preparation classes, etc, are available, and if you have the time and money, certainly won't do you any harm. Really though, knowing how, what and when to adapt in different contexts comes with experience. Most employers recognise this and don't require you to have an additional qualification.
One more point worth mentioning about business English - many teachers worry that they don't know anything about IT, or nuclear physics, or marketing, or whatever field their students work in. Remember that you're not there to teach your students how to do their job - you're there to help them do their job in English. I remember a student who needed help writing presentations that he later had to give to a group of colleagues. He would tell me the subject and the context of his presentation, then sit there, pen poised, waiting for me to tell him what to write! I had to remind him (many times!) that I couldn't come up with the ideas - writing the presentation was his job - I was there to help him with the English he needed to express his ideas.
Anyway, I hope that's of some help...!