I don't think Fernandinho did much wrong, but I do think yesterday's game didn't suit him. You may disagree with my assessment below.
The problem we had was Southampton pressed us extremely well. And as a team we didn't manage it effectively. When we're pressed really hard, the key (specifically in our team) is to be extremely patient, have faith in our passing ability and not be rushed into playing the ball forwards too soon. We didn't do that yesterday. Now the whole team is accountable for that, but Rodri specifically is very good at controlling tempo and helping keep a lid on things. Not always, but normally in the few games he plays where we're overwhelmed initially, he's the one to help turn things around.
Fernandinho didn't do that for us. Now that's not what he's ever done really; historically David Silva has been that guy. But since Silva's left that responsibility has fallen to Rodri, and by proxy when there's no Rodri it's Fernandinho. So since Fernandinho didn't help control play enough, the game became stretched and the team suffered. He only played 58 passes yesterday, which is a particularly low number for a City DM (Rodri normally hits 80 as a minimum for example).
Fernandinho can control tempo on occasion against strong pressing sides, but he needs more support, as Bernardo has notably given him in key games against Liverpool previously.
I'm not blaming Fernandinho for much of this anyhow, I don't think Pep anticipated the way Southampton would play and Fernandinho ended up having to do something he doesn't do. It's like me blaming Aguero for not playing like Jesus when it was needed or vice versa. Rodri may have struggled yesterday too for all I know, but it wouldn't have been for the same reasons. He has his own strengths and weaknesses.
He still did the things he normally does well well. His positioning was solid, he tracked back as efficiently as he ever did, he made a key interception which led to a goal, he was feisty as always, he injected much needed pace into our buildups (although sometimes it was not needed).
He did get tired towards the end. He made a couple risky touches and tackles that are uncharacteristic of him. It was a very end to end game though so I won't be quick to label him as too old off the basis of it.