Use the Picker control which is described here.
You can populate directly in XAML with string or integer values (no elaborate model and binding would required since all you want to capture one of 4 values). The only binding you may need is the SelectedItem property of Picker which provides user selected value.
The link provided gives more help with explanations.
LATER: Re-reading your question I believe you want to do with the TimePicker control itself. I doubt it much if TimePicker has a cutomizable granularity for time increments.
The Telerik version has added properies for customizable increments as you describe, but it is an added value perhaps with licensing implications.