The most immediate action Ivan Jakovlevitch does to show his propriety is to put on a coat. In this instance, we need to assume the coat is not a jacket you wear outside to ward off the cold, but rather part of a suit. By putting on the coat, he is dressing in a more proper and formal manner. He is following the social rules of the occasion. Propriety means to follow or conform to the social rules and behaviors of a situation.
I think they are watching a match. They do not have enough money to buy the tickets. So, they are watching the match by climbing on their cycles, because the stadium boundary wall is very high.