You only need to connect to the laptop to download the program into the Arduino. Then you can unplug it and run the project from battery. That's the same whether it is using a breadboard or some other way to connect the components together.
If you particularly want a soldered version then I would normally use stripboard or perfboard rather than trying to make a specific PCB, unless you're making loads of them.
One thing to note...avoid those little rectangular PP3 9V batteries that show up in so many tutorials. They are very weak and useless for running servos, motors or anything that takes any noticeable power. They cause more "it won't work" problems round here than just about anything else.
Steve