I've come to see my favorite online dictionary in my native tongue says that "stagger to one's feet" means "to come up to standing position in staggering way (this is my translation so might not be so good)", but is this true? I thought it means totally opposite and come to sitting position in staggering way. No?
To get to one's feet means to stand up or to get into a standing position.
You can use other verbs in place of get:
He rose to his feet.
He jumped to his feet.
He sprang to his feet.
He struggled to his feet.
Those are just some examples of different ways that someone could stand up (or get to their feet).
If someone staggered to their feet, then they got to their feet—that is, they stood up—but they did it while staggering.