The copper tape should come with a conductive adhesive backing, so it just sticks to the wood on its own. You'll need to overlap it to ensure electrical continuity. If you've got the copper tape with a non-conducting adhesive, then you'll have to add a blob of solder at each joint. But if so, it really is easier to just order a roll of the conductive-backed tape. Apart from having to take the strings and most of the bridge and scratchplate off again, you can do the shielding at any time. It should take around an hour to do (I know because I did it to a Tele of mine yesterday).

The control plate cover is grounded, so you just need to let some of the copper overlap the edges of the control cavity at the two ends (not the sides, as there's not much of an overlap there, and you don't want it to show) to connect the foil to ground. Likewise, the bridge plate should be grounded, so run some of the copper in the bridge pickup rout over the back end of the rout so that it will make contact with the underside of the bridge plate.

The neck pickup rout is the only area where you'll need to run a ground wire to the back of a pot in the control cavity and solder it to the copper in the neck rout. Alternatively, if there's no visible gap between the control plate and the pickguard, you could simply continue with a strip of copper tape running from the control cavity shielding across to the neck pickup rout. You can then also tape all across the underside of the pickguard to act as a large ground plane, which will all help absorb RFI noise.