Understanding scrolling display circuit
You should be using a 32x4 bit memory.  Why that size?  Four bits (one nibble) wide because that is enough to store one digit.  32 nibbles because you want to store two 16-digit messages. 

It's not clear whether you are actually trying to build this circuit or just design it.  If the former, you're not going to find any 4-bit wide EPROMs.  So you need 32x8 instead, and just use the lower bits 0-3 and ignore bits 4-7.

But you won't find any 32 byte EPROMs either.  The smallest I could find with an 8-bit parallel interface was actually a 8Kx8 EEPROM  (T28C64B).  So you just use 32 bytes out of the 8192 by using five address lines instead of 13.

32 nibbles (or bytes) implies five addressing lines (2⁵=32).  Four of the lines should come from your memory address block, fed by the counter.  This will select one of 16 digits, either from the first or second message  And the fifth will come from your DIP switch, to select which message.  So the first message is programmed into bytes 0-15, and the second into bytes 16-31.  The rest of the EEPROM is ignored.

If this is a real project, you will need to design a separate circuit to program the EEPROM with the messages in the first place.

As far as the display goes, assuming the "scan digits" in the diagram is actually four enable lines, one for each of the four digits, then to show a message you would want to clock out nibbles 0,1,2,3 from the EEPROM while selecting digits from right to the left.  Then clocking out nibbles 1,2,3,4 again selecting the four digits right to left.  And so forth.