You can also keep up with someone when running. If I keep up with my friend when we're running together, that means that I can run at the same speed as my friend.
Yes, sometimes when I spend the night at my friends' house, they keep me up late (cause me to go to sleep later than I should). It is similar to 'stay up'. To stay up is to stay awake instead of sleeping. To keep someone up is to cause them to stay up.
I hope that helps!