It was Ryan at the end, but the story plays out more like Upham learning to fight in real combat situations.
Private Ryan, despite his rescue driving the plot, is only at the very end, and his only contribution is being rescued.
If the story was about Ryan, the movie would focus on him, without the now famous Omaha Beach scene, the squad's trip through Normandy to find him, or anything before the battle at Ramelle, such as the machine gun nest, the sniper in the clock tower, or the German troops in the building next to the Americans.
If it were truly his story, it would have started with the air drop, Ryan being picked up to guard the bridge at Ramelle, and all the inaction that happened between then and the German attack on Ramelle.
Upham, however, starts out a clerk and cartographer from the 29th Infantry Division, so inexperienced that he almost took someone's souvenir Stahlhelm instead of his own M1 helmet, asks if could take his typewriter, and tries to take more equipment than needed. After his breakdown at Ramelle, he begins to fight effectively, and ends up being one of the three in the squad (including Ryan) to survive to the end.