My e-mail on my iPhone randomly stopped updating 2 weeks ago, what can I do about this?

Do you see any error on the screen? If you look down at the very bottom of the page when you swipe to update, do you see a message there? Actually Automatic is not a good setting for it to be on. You should set it to Fetch, and then in Fetch set a time schedule to check mail. Automatic only works when the iPhone is connected to power.

Don't set it to manual, that way you will only pick up mail when you open the Mail app. Set it to something dependent on how much mail you receive and how important it is. If you receive mail sparsely during the day, then set it at longer intervals, it will use less data and battery. If you need to rely on mail more, set it more often, so you are getting your mail at opportune moments. It sounds like you received mail on wi-fi, but missed when you were on cellular data. Check to ensure you have cellular data turned on for Mail.