Specific name of this faucet / sink type
I don't know the name of this type of sink or faucet configuration.  

If you are already expecting to re-use the cap, you may be able to find another faucet that will work with it.  There are a number of one-handle kitchen faucets that are now designed to be installed with or without a wide base.  For example, the Moen Essie faucet 



You may be able to take a faucet like this and mount it on top of the existing cap.