Lay your rolled up carpet runner with the end needing capping extending over a work area for cutting. Measure the width of the carpet runner. Cut off a section of the runner from the end as long as the width of the runner. Measure with a measuring tape on the back of the runner on either side using a black marker. Connect the markings with a straightedge and cut through with a utility knife.
Cut the side sections that contain the pattern off the cut portion. Allow 1/4-inch on the insides of the pattern, cutting firmly with a utility knife and a straightedge. These will be the basis for the end cap.
Place the side section over the top side of the end of the carpet runner. The pattern will be perpendicular between the side section and the main runner, not suitable for a clean finish. Locate the top inside edge of the pattern where it intersects the pattern on the cut section. Make a mark on the back of the cut section. Run a straightedge from the top inside edge of the pattern to the bottom outside edge of the pattern where they intersect the carpet runner pattern. Mark the bottom point on the back of the cut portion. Cut along the line between the marks. Repeat for the opposite side.
Glue the cut edges of the cap portion with white glue. Rub the glue along the cut edge of the carpet backing, taking care not to get the glue on the threads.
Turn the runner over so the backside is facing up. Lay the cut portion over the end of the carpet, also with its back facing up. Trace the outline of the cut portion on the runner. Cut out the traced portion from the runner. Clean up the edges to cut away any loose threads.
Fit the cut portion into the cutout on the runner, with both sections still backwards. Cut sections of carpet seam tape to match the seams between the sections, allowing the carpet tape to overlap where the seams intersect. Trim the outside corners of the seam tape to within 1/4-inch of the outer edge of the runner.
Turn the carpet runner and the capping portion back upright. Lay the carpet tape halfway under the seams of the runner, leaving the other half exposed without the cap portion in place. Hot glue the seam tape to the runner portion of the rug, working one seam at a time.
Place the cap section into the opening, lining up the edges. Hold it in place and lift one half. Hot glue along the seam tape still exposed and press the carpet into the glue, still keeping the seams lined up. Lift up the second half and hot glue the seam tape. Drop the second half in place, pressing hard to make good contact between the rug and the glue.
Turn the capped carpet runner over and hot glue any loose sections of the carpet seam tape.
Massage the threads of the carpet along the seams to form a more unified look, hiding the seam.