Not sure about your problem, but I think you have missed the <TextBox.Text>
tag in your code.
<TextBox x:Name="Name" Grid.Column="4" Margin="0,50,0,0" Grid.Row="2" Style="{StaticResource tooltipError}">
<TextBox.Text>
<Binding ElementName="textBlock" Path="Text">
<Binding.ValidationRules>
<viewModel:MyValidationsRules ValidatesOnTargetUpdated="True"/>
</Binding.ValidationRules>
</Binding>
</TextBox.Text>
</TextBox>