carry out 4 to 16 MUX
I think the OP meant to say demultiplexer.
Yes, is possible with an Uno. No, I don't think a switch case would be suitable.
I would suggest holding the 16 output pin numbers in an array. Also hold the 4 input pin numbers in a second array. Read the 4 input pins and turn those readings into a number between 0 and 15 using bitWrite(). This number can then be used to look up the output pin number from the array.