Will Chief Justice Roberts miss any Court time during a Senate impeachment trial?
Probably not, because according to Senate rules, proceedings will likely begin at 1:00 p.m.
President Clinton's impeachment trial began in the Senate on January 7, 1999, and he was acquitted on February 12, 1999. During this time, Chief Justice Rehnquist presided over all impeachments proceedings in the Senate. But he also attended to his work at the Supreme Court. During this period, 11 cases were argued. Rehnquist participated in each argument, including Davis v. Monroe County Board of Education, California Dental Association v. Federal Trade Commission, and Saenz v. Roe.
How did Rehnquist manage to pull double duty? Fortunately, there were no schedule conflicts. On each argument day, the Court heard two cases in the morning, roughly from 10 a.m. till noon. (I do not know if any afternoon sessions were scheduled). Nor did Rehnquist did not receive any extra staff. (Though Neal Katyal did apply to serve as "a part or full-time law clerk to assist [Rehnquist] with any matters related to impeachment.")
Senate Rule XIII states that "The hour of the day at which the Senate shall sit upon the trial of an impeachment shall be (unless otherwise ordered) 12 o'clock m." Though my understanding is that the Senate would not begin proceedings until 1:00 p.m. (A reporter I spoke with confirmed this understanding with Leader McConnell's office.)
So long as Chief Justice Roberts promptly wraps up any arguments by noon, he should have time to cross the street and get ready to preside over the Senate trial.