I have the following method:
private void DoSomething(CoolClass coolClass)
{
    if (coolClass == null)
    {
        throw new ArgumentNullException(\"coolC         
        By changing following project properties I could eliminate getting this exception while running.
Right click on project -> Properties -> Code Contract (Tab) change the assembley mode to "Standard Contract Requires" also select checkbox - Perform Runtime contract checking
Why don't you just write your own version of the method, if you like the simplicity?
public class CustomContract
{
    public static void Requires<TException>( bool Predicate, string Message )
        where TException : Exception, new()
    {
       if ( !Predicate )
       {
          Debug.WriteLine( Message );
          throw new TException();
       }
    }
}  
Using Code Contracts just to have a friendly API sounds like shooting sparrows with a cannon.