due to my knowlege it is used in common at urban areas through out the states. It is in reference of staying financially prepared, doing what it takes to save or conserve for a particular need/ want. ex. im sticking to my grind, i need to grindup more cash for the holday - its more of a slang word but can be applied in different ways. i apply slang occasionally as a native english speaker but hope i grinded you a decent answer

what does "stay up on" mean in this case?

This can have several meanings. You need to tell us where you heard this phrase.