Don't confuse the adjective arch with the noun and verb meanings related to curves. The latter derive from the Latin word arcus, or "bow." The adjective, on the other hand, comes from Greek arkhos, and it was originally used in compound words like arch-rogue, meaning "the chief rogue." By the 19th century, arch was regularly used as a stand-alone adjective meaning "cheeky" or "forward."
so extremely old as seeming to belong to an earlier period
This meaning relating to being old, ancient, or primitive also comes from the Greek arkhein meaning "to be first." Note the sense of primitive as being "first."