Eye of rump timing questions

cook kit to 125 IT. Can't help you with time as you did not provide enough info.

Rumps tend to have very little fat content and can turn out tough if you're not careful, Bludawg put you in the right direction though.
When I did mine I prepped it with a little cooking oil, kosher salt, coarse black pepper and granulated garlic. I seared it first over direct heat to get a little color and grill marks on the exterior, then I cooked it over indirect heat to an IT of about 125 F and then rested it for about an hour before slicing (you may want to wrap it in foil after 30 minutes of rest time if you aren't planning on eating right away), the carry over while resting should bring the IT to about 130 to 135 and give you a nice medium rare. :-D

you did not say how much this cut weighs so your original Question as to how long is hard to answerer. I'd figure about 30 min a LB but remember each one will cook a little different.