I had some issues with the MouseDown part of this, but here is some code that might get your started.
<Window x:Class="WpfApplication1.Window1"
xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
Title="Window1" Height="300" Width="300">
<Grid>
<Control VerticalAlignment="Top">
<Control.Template>
<ControlTemplate>
<StackPanel>
<TextBox x:Name="MyText"></TextBox>
<Popup x:Name="Popup" PopupAnimation="Fade" VerticalAlignment="Top">
<Border Background="Red">
<TextBlock>Test Popup Content</TextBlock>
</Border>
</Popup>
</StackPanel>
<ControlTemplate.Triggers>
<EventTrigger RoutedEvent="UIElement.MouseEnter" SourceName="MyText">
<BeginStoryboard>
<Storyboard>
<BooleanAnimationUsingKeyFrames Storyboard.TargetName="Popup" Storyboard.TargetProperty="(Popup.IsOpen)">
<DiscreteBooleanKeyFrame KeyTime="00:00:00" Value="True"/>
</BooleanAnimationUsingKeyFrames>
</Storyboard>
</BeginStoryboard>
</EventTrigger>
<EventTrigger RoutedEvent="UIElement.MouseLeave" SourceName="MyText">
<BeginStoryboard>
<Storyboard>
<BooleanAnimationUsingKeyFrames Storyboard.TargetName="Popup" Storyboard.TargetProperty="(Popup.IsOpen)">
<DiscreteBooleanKeyFrame KeyTime="00:00:00" Value="False"/>
</BooleanAnimationUsingKeyFrames>
</Storyboard>
</BeginStoryboard>
</EventTrigger>
</ControlTemplate.Triggers>
</ControlTemplate>
</Control.Template>
</Control>
</Grid>
</Window>