I took a look at your sketch. Which DS1307 library are you using?
As well as changing the variables that represent unchanging data (e.g. pin numbers) to const, another easy win is to replace Serial.println("text") in setup() by Serial.println(F("text")).
I was unable to find monthShortStr(), but I suspect the month strings are another candidate for moving into PROGMEM.
Table divider[12] occupies RAM but appears to be unused.
Variable tempString should be local to the function in which it is used, like timeString. Similarly for tempChar.