1:13 What palace is Paul referring to here?

Paul is most probably referring to Caesar's palace here (cp Php 4:22).

The general consensus among bible scholars is that this letter was written to the Philippians from Rome where the emperor had his palace.

Paul was in prison in Rome but had the opportunity to preach the gospel, and many got saved as a result (cp Php 1:7, 12 with Ac 28:16, 23-24, 30-31).