I had a similar problem: The database server using burstable compute (Standard_B1ms) would suddenly start using up all CPU credits and lock up. Only restarting it would (temporarily) fix the problem - until it locked up again a few days or weeks later.
I've since changed some server parameters and haven't had the problem since. I'm not sure which of the parameter changes did the trick (or if something unrelated changed), but here's a screenshot of my parameters in case it helps someone: