Can i check monotonicity with the marginal utility? For example I have a Cobb-Douglas utility function $U(x,y)$ and I want to check the monotonicity property. 
Can I use the marginal utility functions to see that they are always positive to conclude that $U(x,y)$ has the monotonicity property?
Yes. But you should review the definition of monotonicity (and perhaps of marginal utility) so that you understand why.