Adding two Integers in an Array
Code quality has 3 priorities:


the code works as expected
the code is easy to read and does not surprise the reader
the code is (almost) free from duplications


Please read the following with this 3 priorities in mind.

Naming

Finding good names is the hardest part in software development so always take your time to improve the names of your identifiers.

Single letter names

Do not use single letter names or abbreviations. There are only few exceptions like loop variables or very common  abbreviations. While reading this code your brain has to do additional work to resolve single letters to their meaning. 

Always use expressive names for your identifiers taken from the problem domain, not from the technical solution.

Naming Conventions

Please keep to the Java Naming Conventions. This will make the life of your coworkers easier as it does for you when your coworkers follow this suggestion.

eg.:
You have this method:


public static boolean Continue()



In Java only class names start with an uppercase letter. Also the method name (if correctly spelled) would collide with a reserved Java key word (which is no technical problem since java does distinguish between keywords and method names). This may confuse the reader.

Also since the method returns a boolean its name should start with is or has.

Names should express intention

Your method Continue() reveals another problem: it does not what its name implies.

What it really does is to ask the user what to do. 
So its name should be ie.:  isUserRepeating() 

Comments

Comments should always describe why the code is like it is. 

Your comments somehow structure the code into individual sections.
You should better extract those individual sections to separate methods with names derived from the comments you wrote.

When doing so this will support a very important principle of object oriented programming: separation of concerns.

Visibility scopes

All your variables are declared at the top of your 
main method or the top of your main loop.

You should reduce the visibility to the least possible block which means that you should declare them at first use. This will help to refactor your code later.

Code duplications

You have lots of duplicated code which does merrily the same but  the names of the variables differ.
eg.:


 //Placing the two integer numbers into array
 while(m &gt; 0){
     g = m % 10;
     m = m / 10;
     num1[a] = g;
     temp[a] = g;
     a++;
 }
 while(n &gt; 0){
     g = n % 10;
     n = n / 10;
     num2[b] = g;
     b++;
 }



The only difference is that the first loop additionally sets the values in the temp array.

So the better approach would be to fill the temp array in its own loop (for now, but there are even better approaches):

 //Placing the two integer numbers into array
 while(m &gt; 0){
     g = m % 10;
     m = m / 10;
     num1[a] = g;
     a++;
 }
 a=0; // reset the currentDigitIndex
 while(m &gt; 0){
     g = m % 10;
     m = m / 10;
     temp[a] = g;
     a++;
 }
 while(n &gt; 0){
     g = n % 10;
     n = n / 10;
     num2[b] = g;
     b++;
 }


now you can create a method that does the loop:

private static int placeIntegerIntoArray(int numberToSplit, Integer[] individualDigits){
   int currentDigitIndex=0;
   while(numberToSplit &gt; 0){
         int currentDigit = numberToSplit % 10;
         numberToSplit = numberToSplit / 10;
         individualDigits[currentDigitIndex] = currentDigit;
         currentDigitIndex++;
     }
     return currentDigitIndex;
}


and in your main you change to:

a = placeIntegerIntoArray(m, num1);
placeIntegerIntoArray(m, temp); // initialite to same as num1
b = placeIntegerIntoArray(n, num2);


BTW: this removes variable g from the scope of the main method.

Magic numbers.

Your code uses some literal numbers. You should extract them to constants so that you can give them meaningful names.

eg.: the literal number 10 in


 sum[i] = sum[i] % 10;



might be the SINGLE_DIGIT_MAX_VALUE

whereas in 


 Integer[]  num1 = new Integer[10];



it might be the MAXIMUM_DIGIT_LENGTH