How might I write a eulogy on King Duncan, when he is only in the play for a short time?
He isn't in the play long, is he? But I think you can find some good material on him. Read the scenes he's in in Act 1 (particularly Act 1, Scene 2), and the fact that he is naturally trusting of people, and that will give you somewhere to start. Notice that he's mild-mannered to Macbeth and Lady M as he arrives at their house - and, of course, he gives Lady M a diamond (see Act 2, Scene 1). Not only is he morally nice, he's generous.
That should get you started. But Macbeth also helps you out:
Besides, this Duncan
Hath borne his faculties so meek, hath been
So clear in his great office, that his virtues
Will plead like angels trumpet-tongued against
The deep damnation of his taking-off,
And pity, like a naked new-born babe,
Striding the blast, or heaven's cherubin horsed
Upon the sightless couriers of the air,
Shall blow the horrid deed in every eye,
That tears shall drown the wind.
Duncan is meek, mild, and has been morally "clear" as a king. All of his virtues are so evident that they will be like angels if he is murdered: so much so that people's pity will mean that everyone will be crying, so loud that their tears will drown the wind.
He's clearly, Macbeth thinks, a pretty good king!