Hi Trivia. What to expect, and what to do, about the bore depends on whether your engine is a "normal" B&S or an "I/C" version. The normal B&S generally has an aluminium bore, while the I/C (Industrial/Commercial) has a cast iron sleeve. B&S recommends that aluminium bores should not be de-glazed, but cast iron ones should be. However the first question, which can be answered without taking the piston out, is whether the bore is smooth, or has grooves in it parallel to the piston motion. If it has grooves, it has to be honed, aluminium or not, or the new rings will not seal effectively.
Usually Briggs engines wear out their rings in less time than it takes to produce much of a ridge at the top, but more importantly, mower rings have rather low tension compared with car rings, and there is no real difficulty caused by a small ridge if there is one. With a car engine there are two reasons to remove the ridge. The first is that you won't get the piston out without damaging the ring lands, until the ridge is removed. The rings just will not cross the ridge. The second reason is that if you put new rings in with the ridge still there, the un-worn rings will hit the ridge quite hard at the top of the stroke, and this will damage the piston's ring lands. Neither of these issues is likely to arise with the puny rings in a mower engine, even a Honda with its chromed rings and cast iron bore. (The engines you need to be most careful about are diesels, which have very high ring tensions and very rigid pistons. However there are many other things as well that require really careful work, on diesels.)
As a general rule, a cast iron bore that is being re-ringed should not be honed, but merely de-glazed. I usually do that with some fine emery cloth pressed lightly against the bore with my finger tips, and moved in a circular motion until the shine of the bore is reduced. Then, of course, you must carefully remove all abrasive and dust from the interior. Remember, you must wrap clean rag around the crankpin before you start, to protect it from abrasives and dust. Note that various people have various ways of de-glazing bores, and I'm not about to say that any of them are wrong, especially without seeing the results.
