What is the right course of action here?

You are correct: you may go first.

This is an uncontrolled intersection (no priority signs or traffic lights). In that case, the basic German rule is “right before left” (Rechts vor Links). You are coming from the bottom road and want to go straight. The red car is coming from your left and wants to turn right. Because it is coming from your left, it must yield to traffic coming from its right, which is you. So you have priority over the red car.

The cyclist is on the left side road and is also coming from your left. They would only have priority over you if they were coming from your right (or if there were special signs/traffic lights/crossing rules). Since that’s not the case here, you do not have to let the cyclist go first either.

Rule of thumb for similar questions: At intersections without signs/lights, first check who is coming from your right. If nobody relevant is coming from your right, you generally have priority.