Frankly, most orgs will not reward pro-activity, in fact you can even be punished for it, since if a problem is not yet known by stakeholders then why are you solving it.
There have been times in my career where I spotted issues from other teams on preview or staging servers and helped to fix it. Then I later got blamed (or dragged into the subject) when a similar issue occured on live which I had no association with.
It's better to just sit back and do the minimum, but do it well and professionally. Most importantly, don't make yourself too available:
> Don't respond immediately to messages and emails.
> Don't propose solutions, that you will have to own (at least partly).
> Don't answer questions outside of your responsibility space - even if you know the answer. Instead, direct people to others who should be answering those questions.