Does everyone only have one hour, or am I being grounded or something? This makes a difference, because expectations of electronic communication would change if everyone knew that everyone else only had one hour a week.
If everyone had only one hour, I would download my emails, read them, compose a reply and then upload and send the next time. But if everyone were doing this then there would be no expectation of an e-mail being read or replied to quickly. It really would be like the mail, but electronic. Other than email, I think that a one hour limit would revert the internet to its infant state. That is, I would not use it for social networking, or entertainment at all. I would, through out the week, compile a prioritized list of things I want to find, and the queries which I wanted to use to find them. Then during my hour I would make them, downloading the relevant results to read offline. For example: I need recipes for apple pies, I am curious about mathematical hill climbing, I want to know the lyrics to that song I keep hearing and I would like the headlines from an international newspaper.
I think it is actually an interesting question, how would the structure of the internet change if we were limited in our access. The internet is an economy of attention. Money is made through page views, and clicks, and the value of that attention would shift drastically if we, as users, couldn't afford to get distracted.
On the other hand, if it were just me with the limit, I would just move somewhere pretty with no access at all.