You can create a style to be applied to the Popup in the following way:
<Style x:Key="TooltipPopupStyle" TargetType="Popup">
<Style.Triggers>
<DataTrigger Binding="{Binding PlacementTarget.IsMouseOver, RelativeSource={RelativeSource Self}}" Value="True">
<DataTrigger.EnterActions>
<BeginStoryboard x:Name="OpenPopupStoryBoard" >
<Storyboard>
<BooleanAnimationUsingKeyFrames Storyboard.TargetProperty="IsOpen" FillBehavior="HoldEnd">
<DiscreteBooleanKeyFrame KeyTime="0:0:0.25" Value="True"/>
</BooleanAnimationUsingKeyFrames>
</Storyboard>
</BeginStoryboard>
</DataTrigger.EnterActions>
<DataTrigger.ExitActions>
<PauseStoryboard BeginStoryboardName="OpenPopupStoryBoard"/>
<BeginStoryboard x:Name="ClosePopupStoryBoard">
<Storyboard>
<BooleanAnimationUsingKeyFrames Storyboard.TargetProperty="IsOpen" FillBehavior="HoldEnd">
<DiscreteBooleanKeyFrame KeyTime="0:0:1" Value="False"/>
</BooleanAnimationUsingKeyFrames>
</Storyboard>
</BeginStoryboard>
</DataTrigger.ExitActions>
</DataTrigger>
<Trigger Property="IsMouseOver" Value="True">
<Trigger.EnterActions>
<PauseStoryboard BeginStoryboardName="ClosePopupStoryBoard" />
</Trigger.EnterActions>
<Trigger.ExitActions>
<PauseStoryboard BeginStoryboardName="OpenPopupStoryBoard"/>
<ResumeStoryboard BeginStoryboardName="ClosePopupStoryBoard" />
</Trigger.ExitActions>
</Trigger>
</Style.Triggers>
</Style>
Then, whenever you want to use it, you would write markup similar to this (notice the binding for the PlacementTarget):
<TextBlock x:Name="TargetControl" Text="Hover over me!" />
<Popup PlacementTarget="{Binding ElementName=TargetControl}" Style="{StaticResource TooltipPopupStyle}">
<Border BorderBrush="Red" BorderThickness="1" Background="White">
<TextBlock Text="This is a Popup behaving somewhat like the tooltip!" Margin="10" />
</Border>
</Popup>