You can find his wanted poster on a bulletin board inside Strawberry’s sheriff’s office. Inspect it to start the bounty mission.
The target is located inside an abandoned mine in the mountains of Mt. Shann, in Big Valley. Enter the tunnel for a short cutscene where Arthur checks the camp. As he turns around, the target will be waiting by the entrance. Shortly after, you'll be in duel mode against him. Shoot his hand to disarm him and gain a lot of honor in return.
He will attempt to escape but will be too injured to get away far fast enough. Nevertheless chase after him and use your lasso to stop him and tie him down. Load him to your horse and head back to town.
Along the way, you’ll encounter a group of bounty hunters after Joshua. Kill all of them then loot their bodies. Continue your trip back to Strawberry and place Joshua inside the jail cell to complete the task. Get your reward from the sheriff’s table.