The cat flu was caught from one of Fred's many girlfriends.
He says, "I think I overdid it". Perhaps, "it" refers to snogging female cats— he realised he took a step too far when he snogged a sick one. Now, let's hope she didn't die!The cat seen at the end of the cartoon was Fred's brother.
...Who belongs to a neighbour. He may also become famous (but hopefully he's been neutered and so will remain celibate and not go anywhere near a sick cat.) or maybe he'll be a normal cat but be bugged by conspiracy theorist cats who think he's Fred.