Thats the bad news. And the steering tube should be checked for size and reamed for the bearing cups being installed. The top and bottom tube faces are normally faced at the same time.

In low end bikes where tolerences are wider (read less precision) the cups are often just pressed into the tubes without either of those operations. Those headsets aren't built to the same tolerence levels as more expensive ones<, but are in line with the build quality of the rest of the bike.

But based on your questions, I'd suggest you take the frame to a shop and have it measured and the headset installed there. Not all 1in headsets are actually the same size. You need bearing presses to do this properly and they won't charge a lot. believe me - it'll save you a headache.