When was Arabic written for the first time?

This is a question that has more to do with what you consider to be Arabic than it does with written evidence. We have inscriptions of Paleo-Arabic (also called Pre-Classical Arabic or Old Arabic) which is written with a different alphabet and has some differences in Words) from the first century B.C.E. or the first century C.E. We have writings that are clearly Classical Arabic by the fifth century C.E. What we have in between are a number of intermediates that bridge the incredibly small distance from Paleo-Arabic to Classical Arabic.

Just to elaborate. It is not clear when Classical Arabic first formed since the beginning is an arbitrary point. Language changes slowly over time and while each speaker raises someone who speaks the same language, individual Words and writing styles enter and leave the language with the passage of time. Only when there are too many alterations to prevent intelligibility does it become a distinct language, but each person understands a different part of the whole language. This is why some people understand Shakespeare today, but others do not. Therefore, some would say that Shakespeare is speaking a different language than Current English and others would say that he is speaking the same language (because they understand him).