What kind of meat is used in cheese steak?
There is no single cut of meat that is universally used in Philadelphia.  Top round is common, and it may actually be the most traditional given the sandwich's Italian origins (top round is what is used in braciole and Italian beef sandwiches).  It is becoming more and more common to use rib-eye, though, which is what is used in some of the most popular Philadelphian establishments (e.g., John's Roast Pork and Tony Luke's), which makes sense given its higher fat content.

As lazoDev mentioned, you should be able to get your butcher to slice the meat into thin (~1mm) pieces.  If that is not possible, though, you can put an entire rib-eye roast into the freezer for a half hour or so.  This will harden the meat without actually freezing it solid, and will make thinly slicing by hand much easier.