Hi Zak,
Both the single and double air filter have one housing that doesn't come apart.
I usually use the little pick tool (shown in the linked post) or a small screwdriver to poke as much of the old filter foam out using all the holes in the housing. Compressed air can help with the remaining bits, I usually give it a wash with petrol/gas after.
The new foam is then squeezed in through the opening at the back and again poked through the holes at the front to spread it out to fill the housing.
Your carb will probably need to be stripped and cleaned following the procedure at the start of this thread, Old fuel/oil mix tends to gum up all the parts. Take your time, nothing should fly out, but have a tray underneath to catch any small parts if you want.
A new primer diaphragm gasket is usually required for the engine to run at it's best, the old ones usually dry out & crack (Wallfish or Webhead on here have new diaphragms, as well as the check valve gasket if you need them).
The screw with the spring of the left of the carb is the idle stop adjustment screw, the holes underneath are used on some tools for the throttle cable.
The small quadrant control on the other side of the carb is used to adjust the governed speed. There are two types of this control, one can be set to any position and the other is moved by lifting the spring loaded washer & has five positions. Early production carbs don't have this control.
David