I've messed around with archery a few times from compound bows to replicas of English longbows. I'm well aware there's a difference between effective range, target range, and maximum distance. A marked shot of 500+ yards seems absurd to me when a compound bow has a maximum range of around 100 yards. The longest aimed shot of an English longbow was recorded (according to Wikipedia) in the 16th century and was 345 yards. Surely this is a gross exaggeration? Has anyone come even remotely close using a replica? How would a Mongol bow compare to a Magyar bow, as they seem to have similar construction and were used in roughly this same time period (late 12th to early 13th century)?
From a numbers standpoint, it seems exaggerated. English longbows had a draw strength at around 700 N and had "less than 300" feet per second initial velocities. At 500 yards exactly, this means the arrow would drop ~60 meters and have a flight time of just over 6 seconds when accounting for drag with a 53.6 gram projectile (replica longbow arrow.)
Wind would further screw up the chances of hitting a target, and the way the story is told, sounds like the feat was accomplished with at most a few tries.
Furthermore, the maximum distance someone has shot with an English longbow under controlled circumstances is 412 meters by József Mónus, a Hungarian master archer.
While the feat is plausible depending on how different the Mongols liked their bows, it seems exaggerated.