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Using speed measurements of the gas at different distances from the Galactic center, the Sun appears to be cruising along at 200 kilometers per second and it takes 240 million years to complete the grand circuit around the Galaxy. This speed is an absolute speed.
The Milky Way does not sit still, but is constantly rotating. As such, the arms are moving through space. The sun and the solar system travel with them. The solar system travels at an average speed of 515,000 miles per hour (828,000 kilometers per hour). Even at this rapid speed, the solar system would take about 230 million years to travel all the way around the Milky Way.