I believe a number of third party calendar apps in the App Store will have a monthly view similar to that. For example, Calendars (or Calendars 5) by Readdle will offer something similar looking for monthly view.
As for being able to edit a calendar online, well, you can use a calendar from a provider that has an online presence where you can login and view and edit your calendar. I believe Apple's iCloud allows for that, Google's Calendar, Microsoft's Outlook, etc.