Some factors you haven't detailed that could contribute are weather and terrain but I am going to gloss over them as well. Other factors I will gloss over are the differences between a strong and a weekweak horse, the skill of the rider and how the horse was ridden. Some of these complexity's are explained/addressed here: Possible distance travelled by horse over 6 weeks?
So an oversimplified estimation would be:
Approximate distance to travel: 452 km or 280 miles
Approximate distance a moderately trained courier horse can travel in a day: 52 km or 32 miles
Therefore, from Byzantium to Ancyra would take approximately 9 days. So, round trip, assuming he stayed the night in Ancyra its going to take about 18 days.