Acts 13:13-16
First, notice that Paul does this on the Sabbath day in a synagogue. He speaks to "brethren." In this context, brethrendoes not mean fellow church members but "fellow Israelites"—Jews, Levites, Benjamites, or whatever tribes they were from. They were brethren in terms of their ethnic relationship.
Second, he also speaks to those described as "you who fear God." This term was used of Gentiles who were proselytes of Judaism, Gentiles who had converted to Judaism. Here is a congregation in a synagogue, consisting of Jews and Gentiles already converted to Judaism—keeping the Sabbath. By the way, this took place in the environs of Galatia.
John W. Ritenbaugh
The Covenants, Grace, and Law (Part Twenty-Four)

Following Christ's Sabbath-keeping example, Paul imitated Him and instructed believers to imitate him in turn. Paul preached to both Jews and Gentiles on the Sabbath, demonstrating that the Sabbath is intended for everyone, not exclusively the Jews. The Gentiles observed the Sabbath alongside the Jews in the synagogue on the seventh day, not on Sunday.