Take a look at the sample application as it has geometry operations under the Geometry section to get an idea of the capabilities.
The answer to your question is to use an intersect.
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So draw your circle using the buffer (red), then define another polygon to intersect the arc segment (green) and call the intersect operation which results in your arc (yellow)
I've done a quick example below. The code is a bit shoddy, but you should get the idea:
private void drawArc(MouseEvent event) {
//centre of arc where we clicked the map
Point centrePt = map.toMapPoint(event.getX(), event.getY());
//buffer to get circle as a polygon
Polygon circle = GeometryEngine.buffer(centrePt, map.getSpatialReference(), 5000000, null);
//create a line of the circle
Polyline circleLine = new Polyline();
circleLine.addPath(circle, 0, false);
//draw the circle
SimpleLineSymbol redLine = new SimpleLineSymbol(Color.RED, 2);
Graphic circleGraphic = new Graphic(circleLine, redLine);
graphicsLayer.addGraphic(circleGraphic);
//create intersection geometry
Polygon intersection = new Polygon();
intersection.startPath(centrePt);
intersection.lineTo(centrePt.getX(), centrePt.getY() + 10000000);
intersection.lineTo(centrePt.getX() + 10000000, centrePt.getY());
//draw intersection geometry
SimpleLineSymbol greenLine = new SimpleLineSymbol(Color.GREEN, 2);
Graphic intersectGraphic = new Graphic(intersection, greenLine);
graphicsLayer.addGraphic(intersectGraphic);
//now we can perform an intersection operation to create the arc
Polyline arc = (Polyline) GeometryEngine.intersect(circleLine, intersection, map.getSpatialReference());
//draw the arc
SimpleLineSymbol yellowLine = new SimpleLineSymbol(Color.YELLOW, 5);
Graphic arcGraphic = new Graphic(arc, yellowLine);
graphicsLayer.addGraphic(arcGraphic);
}