Well, that is one way to approach the problem.
Your equals will probably work provided that you change Object object = (Object)o; to cast to the real class.
However, equals ought to cope with the case where o is not of the expected type. The contract requires you should return false rather than throwing a ClassCastException ...
You would then use list.contains(o) to test if an object with the same int value exists in the list. For example:
if (!list.contains(o)) {
list.add(o);
}
But when you override equals, it is best practice to also override hashcode ... so that your class continues to satisfy the equals / hashcode invariants. (If you neglect to do that, hash-based data structures will break for your class.)
However, this won't scale well, because the contains operation on an ArrayList has to test each element in the list, one at a time. As the list gets longer, the contains call takes longer ... in direct proportion; i.e. O(N) ... using Big O complexity notation.
So it may be better to use a Set implementation of some kind instead on ArrayList. Fepending on which set implementation you choose, you will get complexity of O(1) or O(logN). But the catch is that you will either have to to implement hashcode (for a HashSet or LinkedHashSet), or implement either Comparable or a Comparator (for a TreeSet).