Do I join the pin with the hole next to it on the board? Do I solder straight on the arduino pin? (But, they are really close... feels dangerous, in case the wire sticks out a little etc.)
Personally I use the boards having long continued copper stripes. For all DIL circuits I just cut the stripes under the circuit. That way building mountain ridges, baking, with tinn is avoided.
Most of the time, connectors go at the edge of the board (as your first image shows). In your case, you will use wires to to connect the connector to the Nano pins.
If you use strip board (as mentioned by @Railroader), you can cut the cut the strips at specific places; I usually use a drill bit that I roll between my fingers.
If you look closely at the the two rows of the zif-socket (in the centre), you will see that they were cut so the one row of pins does not connect to the other row of pins; there are probably other "breaks" in the strips but they are not as clear (e.g. for the dip switches.