Static binding doesn't update when resource changes

First of all, your property is actually not a property, but a field. A minimal property declaration would look like this:

public static SolidColorBrush Property { get; set; }

Please note the name is starting with an uppercase letter, which is a widely accepted coding convention in C#.

Because you also want to have a change notification fired whenever the value of the property changes, you need to declare a property-changed event (which for non-static properties is usually done by implementing the INotifyPropertyChanged interface).

For static properties there is a new mechanism in WPF 4.5 (or 4.0?), where you can write a static property changed event and property declaration like this:

public static class AppStyle
{
    public static event PropertyChangedEventHandler StaticPropertyChanged;

    private static void OnStaticPropertyChanged(string propertyName)
    {
        StaticPropertyChanged?.Invoke(null, new PropertyChangedEventArgs(propertyName));
    }

    private static SolidColorBrush property = Brushes.Red; // backing field

    public static SolidColorBrush Property
    {
        get { return property; }
        set
        {
            property = value;
            OnStaticPropertyChanged("Property");
        }
    }

    public static void ChangeTheme()
    {
        Property = Brushes.Blue;
    }
}

The binding to a static property would be written with the property path in parentheses:

Background="{Binding Path=(style:AppStyle.Property)}"