According to the cockpit photos of the of the Electra and Lockheed literature, the 10E had a heading indicator or directional gyro which allowed Earhart to fly a true course. She also had a Sperry gyro-horizon, Pioneer compass, altimeter, airspeed indicator, turn and bank, rate of climb and a Sperry autopilot which held the plane on the course selected. The magnetic compass was a standby compass and had a deviation card which indicated the compass heading to steer a true heading.
When Noonan clocked the sunrise he knew according to his nautical almanac where the dawn line that he was on was located and that the angle of that line was, 157/337, for that day. The line was a true heading plotted on his chart. He knew what the sunrise time at Howland was and therefore he knew how far away they were from the line that ran through Howland.
When, according to his chronometer, they arrived on the LOP looking for Howland, he had Earhart
turn on to a heading of 157 or 337 using the heading indicator which would have been a true heading. To correct for wind, Noonan would figure the wind correction angle and tell Earhart what heading to steer. According to what I have read the wind was most likely out of the NNE, therefore, going down the 157 line they would have had a left quartering tailwind.