The age estimate for B1509 of 1700 years is given in Earth's time-frame, referring to when events are observable at Earth. So, if early astronomers were looking in the right place at the right time they would have seen the supernova explosion that created the pulsar about 1700 years ago (incidentally, since this object is located a long way to the south, Chinese and European astronomers would not have been able to see it). Put another way, in this image B1509 looks about 1700 years old.
Astronomers usually quote ages in Earth's time-frame, since their knowledge of the Universe is based almost entirely on the electromagnetic radiation received by telescopes. Because the speed of light is finite, we don't even know what's happening "now" on the Sun, since light takes about 8 minutes to reach us at the Earth.
The time for certain astronomical events such as supernova explosions can be known very accurately in Earth's time-frame, sometimes within just days or hours or seconds, but very large margins of error can be incurred if the light travel time is included in an "absolute" age estimate, since distances are much harder to measure accurately. Within the Milky Way galaxy, distance errors for objects of hundreds or thousands of light years are common. So, it makes sense for astronomers to use relative times based on what they see in images, rather than absolute times that include the light travel time.