How does earthing affect potential difference in cylindrical capacitors? 
I can't get my head around how earthing the outer cylinder affects the potential difference between the plates. Also, if you positively charge the inner cylinder, then negative charge should accumulate on the inner side of the outer cylinder. So the outer side is at a positive potential. So shouldn't it mention which side of the cylinder we've earthed, because the charge flowing out should depend on that. 
I've solved the problem, but I just need some physical intuition for the P.D between the plates (thank you photon for pointing that out). 
 A: Be careful with the wording of the question. It doesn't ask for the potential difference between the cylinders (although you'll have to calculate that to get the final answer). It asks for the "potential of the inner cylinder".
You can only give the potential of an object if you know what the reference potential is, and can determine the potential difference between the object and the reference. 
By telling you the outer cylinder is earthed, they've given you enough information to do that.

negative charge should accumulate on the inner side of the outer cylinder. So the outer side is at a positive potential.

This is an incorrect inference. We can choose our reference potential however we like, so despite there being charge accumulated on part of an object, the potential of that object could be positive, negative, or zero depending on how we choose our reference.
A: When outer surface is not connected to earth surface then on inner surface of outer cylinder -q is induced and on outer +q on outer surface.When we connect outer cylinder to earth then +q on outer surface flows down to earth(but earth does not get charged as earth is charge sink) so that whole system potential energy decreases.If -q from inner surface of outer cylinder would have flown then potential energy of whole system of arrangement would have increased(which is not possible law of universe potential energy always tends to decrease).Now one might whole Potential energy would decrease if whole charge on outer cylinder would come to earth but this would not be equlibrium state as electric field inside outer cylinder is non zero so again charges would accumulates.So cylinder does not see where wire is connected on inner or outer just goes to equlibrium state with least potential energy.Thus outer cylinder has -q on inner surface due to attraction from +q on inner cylinder and outer cylinder comes to 0 potential as of earth and there exsist some P.D. between cylinders.
