There are two ways to do this.
In this approach, you sort the ArrayList using a custom Comparator. 
Here is the Comparator code. Notice how it first tries to sort on numberType, and then number:
public class PhoneNumberComparator implements Comparator<PhoneNumber> {
    @Override
    public int compareTo (PhoneNumber lhs, PhoneNumber rhs) {
        int result = lhs.numberType.compareTo(rhs.numberType);
        if (result == 0) {
            result = lhs.number.compareTo(rhs.number);
        }
        return result;
    }
}
Then you can call:
Comparator c = new PhoneNumberComparator();
arrayListToSort.sort (c)
Now by itself, this won't work completely because the sort of numberType will just be in string order.  The most convenient way to impose the ordering is to make numberType an enumeration.  Here is the code for that:
public enum NumberType { MOBILE, HOME, OFFICE }
Then the PhoneNumber must be defined so that numberType is a NumberType:
public class PhoneNumber {
    public String number ;
    public NumberType numberType;
    // .......
}
(By the way I would encourage you to also make number and numberType into private variables, and add getter methods, as per the JavaBeans standard.)
Make PhoneNumber Implement Comparable
If you are planning to do this sort often, then instead of writing the Comparator, you should make PhoneNumber implement the Comparable interface:
public class PhoneNumber implements Comparable <PhoneNumber> {
    public String number ;
    public NumberType numberType;
    // .......
    @Override
    public int compareTo (PhoneNumber rhs) {
        int result = this.numberType.compareTo(rhs.numberType);
        if (result == 0) {
            result = this.number.compareTo(rhs.number);
        }
        return result;
    }
}
Then you can just call:
arrayList.sort()
You still need to make NumberType an enum, as discussed in the first approach. The above compareTo() implementation relies on NumberType being Comparable; all enum instances are automatically Comparable.
Notice the similarities and the differences between the Comparator and the compareTo() implementations. A good discussion of Comparator and Comparable can be found here.