You are kind of caught in a "time warp" - what that was designed for versus what today's expectations might be. Those four holes fit a sheet metal bended mount made by Weaver for their 3/4" (?) or 7/8" (?) ".22 scopes". Today we would consider those scopes very dim, not sealed, not particularly robust. But have been used to take millions of gophers (I am sure) out here on the Prairies. I, for one, thought they were a huge step up from the iron sights, at the time.
That Weaver side mount that you mention also fits those holes. But you also need the partner side mount rings - they come in two lengths - then you can mount a 1" scope onto your Cooey - I have done so a few times. I had thought the Cooey 600 had the dovetail grooves along the top of the receiver - maybe I am mistaken. Over the years, some felt that the 3/8" dovetails / 3/8" clamp rings "weren't good enough", (although what is used on Lee Enfield 303 British T0-1 rail) so some chose to drill and tap on top and install conventional bases with conventional 1" rings. If you want to mount a 1" tube scope on your Cooey 600, and if it does have those grooves, just buy the 1" scope rings with the 3/8" clamp size. Probably want to ensure your "new" scope has Adjustable Parallax - a cheap "high power rifle" scope will likely have significant parallax sighting error at rimfire ranges... "Rifle" scopes commonly are set parallax free about 150 yards; "rimfire" scopes often set parallax free at 50 or 60 yards.