If you keep the USB strips adjacent and as short and neat as possible and you'll likely have no problems for a prototype.
For USB low speed, the 1.5Mbps is so low that you don't really need to put any effort into the routing. Even USB full speed at 12Mbps is pretty forgiving.
In terms of cable to board, cables with USB connectors on one end and 0.1" pitch headers on the other end are cheap and readily available, so that would solve trying to connect to the strip board. Alternatively a USB B connector has a pitch and shape that it is possible to install them on strip board.