The short shelf is typical of many older peter wright anvils. I would not repair the anvil unless you find that the flaws in it are hampering your work. When you do, there are several knowledgable people about anvil repair locally who are members of the New England Blacksmiths. (
New England Blacksmiths we're having our spring meet up in brentwood NH the weekend after memorial day ) I know Ed Cannon who lives out by the cape, and John D'Abate who lives in Rhode Island near me both have done multiple anvil repairs on various types of anvils. I believe that Ed uses 11018 rod for his anvil repairs.
You have to do the welding in just the right manner to prevent from pushing heat around and causing delamination of the face, or destroying the overall hardness of the face.