Are circuit boards necessary?
You can literally solder components together, i.e. one end of the resistor to the base of the transistor etc. without using a prototype board, as long as you are careful not to let any leads short together.  Unlike most cases, where you would twist wires together for mechanical strength, when prototyping this way you generally want the solder to hold the leads together so it is easier to later unsolder and reuse the components again.

Instead of soldering, you can also just clip leads together using alligator-clip jumpers.  Either would work for a circuit of the complexity in your question.  Much more complicated, and you will want to get a solderless breadboard (however, see below!).

You can probably stick the other end of the resistor directly into the socket of the Arduino.  You will need to get some hookup wire for the ground and +5V leads; 22-guage solid is probably the best size.

Before PCBs were common, consumer electronics were all hand-wired.  Components with leads were soldered together using terminal strips, like the one at the top of the picture.



This is the chassis of a 1948 Motorola Golden View 7" television set.