I'm starting to get upset with my boyfriend for staying up until 5 AM.
Is it because he wakes you up?
Yes, I get upset when he keeps me up late.
You should tell him to stop if he keeps waking you up. There should be a reasonable solution.
What kind of solution?
It sounds like you need to work out a compromise with him. When you and your partner have conflicting needs, you need to find a middle ground where you can both be satisfied. Maybe, your partner can continue to stay up late but he has to put in extra effort to be quiet.