Not in the sense that Roman law immediately extended to everyone in the Bosporan Kingdom, no. Roman law only applied to Roman citizens:

"Roman law...was primarily ‘personal’ rather than ‘territorial’ in its reach. That is, ‘Roman law’ was not thought of as the law of a particular area, but law for Roman people, wherever they happened to be." (Andrew M. Riggsby, Roman Law and the Legal World of the Romans, pg. 103)

Roman law was originally territorial, sort of, since it was intended to apply within the boundaries of the city of Rome. As Rome expanded its power over its neighbours, the law didn't expand with it. The law still only applied to Romans. Occasionally Roman citizenship was granted to other people (the other Latin-speaking inhabitants of central Italy, eventually all the other people in Italy) and individuals from other areas could become Roman citizens in various ways.

Roman law applied to Romans, or anyone with Roman citizenship rights, no matter where they were, so if a Roman citizen was in the Bosporan Kingdom, then certainly they could appeal to Roman law. But anyone else in the kingdom didn’t immediately gain Roman citizenship just because the kingdom was a Roman client state/province.

It wasn’t until 212 that most people in the Roman Empire become actual Roman citizens (and therefore subject to Roman law), under the Edict of Caracalla.