Can character reputations be updated during the game based on past events that is going to be defined as the game unfolds? Would that be useful?
That's an excellent idea! Character reputations are intended to help you write your first letters: "I write to you because your fine work in Ireland distinguishes you as a scholar of great merit" and so forth. Once character introductions have been made, the Mystery usually takes precedence.
However, if anything happens during the game that the other Investigators might reasonably hear about without requiring letters to be sent, feel free to let the other players know. If you like, you could create in-world artefacts to communicate this information. For example, if you roll a 79–83, you could write a newspaper article mentioning that your character has been arrested.