Can I solder the components together? I just started to learn and I don't have a circuit board. 
Electronic – Are circuit boards necessary
boardsoldering
Related Solutions
I use a Metcal system. Provided that the correct cartridge is used, I get a perfect joint every time, with minimal risk of damaging parts. Metcal uses RF heating with very accurate temperature control built into the cartridges. They have excellent tutorials on soldering techniques, like this one:
Here's another interesting method of perfboarding, this guy does it with SMD components and really tiny wires. He made a special little wiring pen for laying down the wires:
(source: davr.org)
Personally I make solder bridges for pins that are in adjacent holes, for things further away I use thin wires (though not as thin as the ones above) in adjacent holes to the things I want to connect.
And no, I didn't just wire those pins all on the fly, I had drawn up a detailed schematic in advance. There's no way I could keep track of so many wires just in my head.
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Best Answer
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.