A rectifier is a rectifier - getting the Honda one will get you the exact connector and avoid any hassles there. However, there are quite a few aftermarket ones and you can use an industrial standard 3 phase rectifier bridge block, or two single phase bridge blocks. The wiring is pretty simple to figure out if you decide on a non-OEM type.
The rated voltage issue is important, rectifiers are rated for maximum current (25A should be plenty) and PRV, peak reverse voltage: this is essentially the maximum voltage it will handle. Although the bike is a 12V system, the actual peak reverse voltage at the rectifier will be higher. Using a battery charger can give quite a bit higher voltage, a 20A charger will put as much voltage as it can to try and get 20A flowing. As a rule do not use anything more than a 2A charger for these little batteries, 5A will be OK if that's all you can find in an emergency but if you buy a charger for the bike get 2A max and an automatic "battery tender" bike battery charger is much recommended.
As for the wiring, HM is right that the wiring is pretty simple on a K1, only one fuse and much simpler circuitry than the later multi-fuse models. I can't imagine what your terminal strip (definitely not original!) was added for, you would have to trace the wires to figure it out. If there are original wires attached you can determine what they are by colour, using a Honda owners manual or shop manual wiring diagram, you should be able to find one online for download. Avoid the Clymer and Haynes wiring diagram, they skip stuff and simplify stuff: danged annoying to try and figure out a harness when the diagram is not complete. A diagram for a later model with three fuses will make you nuts, although the basic wire colours are the same basic functions the ignition switch and fuse wiring is complicated and just confusing if your bike is a single fuse model.