That's ok. The answer just differs per platform, since styling is quite platform dependent. The WPF MapView control uses a "Callout" control to render the callout. So you can change the styling completely by using an implicit style.
<esri:MapView>
<esri:MapView.Resources>
<Style TargetType="esri:Callout">
<Setter Property="FontFamily" Value="Courier New" />
<Setter Property="BorderBrush" Value="Red" />
<Setter Property="BorderThickness" Value="5" />
</Style>
</esri:MapView.Resources>
</esri:MapView>
Also note that you can use the ShowCalloutAt overload that takes a UIElement. This allows you to just create a UserControl with whatever content you'd like and get it rendered inside the callout border, instead of relying on the default 4 text, description, button, icon parts.