This mission is a purely time based one if you want the Gold Medal and you’ll have to skip cutscenes if you wish to achieve it, there’s no other way around it. As such, it might be best to replay this mission to go for the Gold Medal or to watch the cutscenes (it will be easier to obtain this Gold on a first run).
Head to the yellow marked “BR” tag by the saloon, the western most of the three of the map. Arthur will meet up with Dutch before heading inside. Make your way to the bar and call on the bartender twice to get served where Arthur will ask about Angelo Bronte. Head outside the Saloon now and through the alley to the right of it, heading through to the end to find a the kids that the drunken man told you about. Follow them until you reach the nearby Church where Cleet will rob Arthur and run off.
This begins the street chase and you’ll need to tackle the kid before he reaches the market if you want the Gold Medal, this can be very difficult indeed. Give chase up the street on foot until Cleet jumps off the wagon and on to the nearby trolley, quickly jump on the nearby horse and give chase some more.
Eventually, Cleet will get off the trolley and head right into alley, give chase and ignore speaking to the people as you dart through. If you’re quick enough, you’ll get your first chance to tackle Cleet here otherwise continue the chase out into the street once more. Avoid the wagon that comes across and ignore the alley that he went down, instead following the wagon to the end of the street and then left and over the wall. As you head up the stairs, Cleet will get slowed down giving you the perfect opportunity to tackle him and get your cash back along with the location of Angelo Bronte.
Make your way back to the saloon and head over to the benches on the other side to find Dutch and end the mission. If you skipped the cutscenes and tackled Cleet before he reached the market then you should obtain the Gold Medal.