Yes, there are some FOSS apps that can help you with lunar cycles and a 13-month, 28-day calendar. Here are a few options:

1. Calendars - Free Calendar & Planner for Android: This app has a feature called "lunar phases" which shows the current phase of the moon along with information about each phase. You can also set up lunar calendars to track the cycles.

2. LunaCal for Android: This is an advanced calendar app that includes a feature called "Luna" that helps you keep track of your menstrual cycle and other aspects of women's health, including fertility and pregnancy. It also has a lunar phase feature.

3. Moon Phase Calculator by Glyphics: This is a web-based tool that allows you to calculate the moon phase for any given date. You can also view the phases in different ways, such as a graph or chart.

4. Dark Sky Weather Forecasting App: This app has an option called "moon data" which provides information about the current and upcoming phases of the moon.

I hope this helps!