As an assistant, it is your job to help keep other people on track, particularly your manager. As the person in charge of scheduling, you are responsible for making sure that you manager arrives at meetings on time. You need to remind your manager about meetings and help keep them on schedule. Reminders can be set through email, given in person, or over the phone. You may also need to extend the reminders to other people attending the meeting.
Project updates will probably be sent to you as well. You should establish timelines for updates, and check in with employees when if the updates do not show up on time. It is not your job to hover over other employees, but staying aware of updates on the timeline will help you keep other employees on track to complete their projects on time.