Here "judgment seat" refers to Festus ruling over Paul's trial as judge. AT: "I sat upon the seat to act as judge" or "I sat down as judge" (See: [[rc://en/ta/man/translate/figs-metonymy]])
To "stand trial" is an idiom meaning to speak to a judge so the judge can decide if a person is right or wrong. AT: "to go to trial about these charges" or "for a judge to decide if these charges against him are true or not" (See: [[rc://en/ta/man/translate/figs-idiom]])