Для изменения значения свойства Prop1 в визуальной части элемента, нужно использовать Binding.
<UserControl x:Class="Primo.SDKSample.PrimoElement"
xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
xmlns:mc="http://schemas.openxmlformats.org/markup-compatibility/2006"
xmlns:d="http://schemas.microsoft.com/expression/blend/2008"
xmlns:local="clr-namespace:Primo.SDKSample"
mc:Ignorable="d"
d:DesignHeight="450" d:DesignWidth="800">
<Grid>
<TextBox Text="{Binding Prop1}" IsReadOnly="False" BorderThickness="0" Grid.Column="0" Margin="5,0,5,0" Height="23" TextWrapping="NoWrap" VerticalAlignment="Center" BorderBrush="#D0D7E2" />
</Grid>
</UserControl>