Since it was asked for, full code (minus loop()):

/* Sketch for Arduino to run the automatic cat feeder.
 *
 * 
 */

//includes
#include <Servo.h> //for running the servo
#include <Wire.h> //required for RTClib
#include "RTClib.h" //to run the real time clock functionality. https://github.com/adafruit/RTClib

//status indicator values
const int onboardLedPin = 13;

//buzzer values
const boolean buzz = true; //turns on and off the buzzer functionality
const int buzzerPin = 8;
const int buzzLength = 250;
const int buzzDefaultFreq = 1000;
const int buzzFeedStartFreq = 2000;
const int buzzFeedEndFreq = 500;
const int buzzBattWarnFreq = 2000;
const int buzzConfirmFreq = 100;

//Servo setup
Servo dispenserServo;
const int servoPin = 11;
const int feedServoSpinTime = 2000;
const int STOP = 90;
const int SPIN_CLOCKWISE = 180;
const int SPIN_COUNTER_CLOCKWISE = 0;

//clock
boolean timingEnabled = true;
RTC_DS3231 rtc;
String feedTimes[] = {"7:00"};

//button
const int buttonPin = 7;
int buttonVal = 0;

//other
const int mainWait = 5000;

/*
 * Does a buzz if it is set to.
 */
void doBuzz(int frequency, boolean wait){
  if(buzz){
    tone(buzzerPin, frequency, buzzLength);
    if(wait){
      delay(buzzLength);
    }
  }
}

/*
 * Default function to not wait for buzz to finish.
 */
void doBuzz(){
  doBuzz(buzzDefaultFreq, false);
}

void checkButton(){
  buttonVal = digitalRead(buttonPin);
}

/*
 * Determines if it is time to do the feeding.
 */
boolean isFoodTime(){
  if(!timingEnabled){
    return false;
  }
  DateTime curTime = rtc.now();
  
  String hr = String(curTime.hour());
  String mn = String(curTime.minute());
  
  String time = hr + ":" + mn;
  
  for(int i = 0; i < sizeof(feedTimes); i++){
    if(feedTimes[i] == time){
      return true;
    }
  }
  
  return false;
}

String formatDateTime(DateTime dtIn){
  return String(dtIn.month()) + "/" + String(dtIn.day()) + "/" + String(dtIn.year()) + " " + String(dtIn.hour()) + ":" + String(dtIn.minute()) + ":" + String(dtIn.second());
}

String formatDateTime(){
    formatDateTime(rtc.now());
}

int getNumFromButton(int startAt){
  doBuzz(buzzFeedEndFreq, true);
  int longPressLength = 500;
  int output = startAt;
  bool counting = true;
  do{
    checkButton();
    if(buttonVal == HIGH){
        delay(longPressLength);
        checkButton();
        if(buttonVal == HIGH){
          counting = false;
        }else{
          output++;
          doBuzz(buzzConfirmFreq, true);
          delay(longPressLength);
        }
    }
  }while(counting);
  doBuzz(buzzConfirmFreq, true);
  delay(buzzLength);
  doBuzz(buzzConfirmFreq, true);
  return output;
}

void setTime(){
  //rtc.adjust(DateTime(F(__DATE__), F(__TIME__)));
  //Serial.print("Setting the clock... ("+formatDateTime()+")\n");
  delay(buzzLength);
  delay(buzzLength);
  doBuzz(buzzConfirmFreq, true);
  delay(buzzLength);
  doBuzz(buzzConfirmFreq, true);
  delay(buzzLength);
  doBuzz(buzzConfirmFreq, true);
  Serial.print("Getting Year... (");
  int year = getNumFromButton(2000);
  Serial.print(String(year) + ")\nGetting Month... (");
  int month = getNumFromButton(1);
  Serial.print(String(month) + ")\nGetting Day... (");
  int day = getNumFromButton(1);
  Serial.print(String(day) + ")\nGetting Hour... (");
  int hour = getNumFromButton(0);
  Serial.print(String(hour) + ")\nGetting Minute... (");
  int minute = getNumFromButton(0);
  Serial.print(String(minute) + ")\nGetting Second... (");
  int second = getNumFromButton(0);
  Serial.print(String(second) + ")\n");
  delay(buzzLength);
  doBuzz(buzzConfirmFreq, true);
  
  Serial.print("Finished getting numbers.\n");
  rtc.adjust(DateTime(year, month, day, hour, minute, second));
  Serial.print("Clock set to: " + formatDateTime() + "\n");
}

/*
 * Setup of things:
 *  - status led
 *  - buzzer
 *  - servo
 */
void setup(){
  Serial.begin(9600);
  Serial.print("\nBeginning setup of the CatFeeder.\n");
  // status led
  pinMode(onboardLedPin, OUTPUT);
  // buzzer
  pinMode(buzzerPin, OUTPUT);
  doBuzz();
  //servo
  dispenserServo.attach(servoPin);
  pinMode(servoPin, OUTPUT);
  dispenserServo.write(STOP);
  
  if(timingEnabled){
    //clock
    boolean rtcOn = rtc.begin();
    Serial.print(String(rtcOn) + "\n");
    if(! rtcOn){
      Serial.print("Couldn't find RTC. Timing Disabled\n");
      timingEnabled = false;
      doBuzz(buzzConfirmFreq, true);
      delay(buzzLength);
      doBuzz(buzzConfirmFreq, true);
      delay(buzzLength);
      doBuzz(buzzConfirmFreq, true);
      delay(buzzLength);    
      doBuzz(buzzConfirmFreq, true);
    }else{
      if(rtc.lostPower()){
        Serial.println("Clock lost power.\n");
        setTime();
      }
    }
  }
  if(!timingEnabled){
    Serial.print("Timing disabled. Will only actuate on button press.\n");
  }else{
    Serial.print("Current time: " + formatDateTime());
  }

  //button    
  pinMode(buttonPin, INPUT);

  Serial.print("Finished setup.\n\n");
}

Full code size:

Binary sketch size: 12,416 bytes (of a 32,256 byte maximum)