To piece in linoleum, securely tape a remnant piece with duct tape, exactly over the existing linoleum. Use a really sharp Exacto knife to cut thru both the old and new pieces over a pattern line if possible, making an exact duplicate replacement.
Remove the old piece of linoleum, and scrape away and excess glue. Apply a small amount of glue, and insert the new piece into the cut out.
Set some heavy blocks, books or wood over the newly installed piece, and let it dry for a day. The next day, remove any oozed glue and paint a thin coat of clear fingernail polish directly over the new seam, and you should be good to go.
This usually will work with good satisfaction unless the new remnant piece and the old linoleum are already discolored, and if this is the case, it's time for a complete replacement.