Each MAC generation is completely different from the previous one, so a direct change is not possible. You'll have plenty of space inside the G4 case, but the displays are not interchangeable; they use a different connector, and the retroillumination is different: G4 has a CCFL display (you'll need to find a way to power the inverter, and you'll lose the ability to regulate the brightness), while Air uses a LED display.
The battery: the connectors are the same, but the data signals are different, so you'll not be able to properly charge the old one.
As for the keyboard/trackpad: internally they are the same, but the wiring is different, so you'll need to modify the connectors.
It seems that the ebay module is capable to adjust brightness, contrast, colour, etc. using the push-button interface. To be sure, please contact the seller for more information. Also, provide him with the exact Powerbook display model, to be sure the adapter will work as intended.
I have to find a way to power then LCD controller either from battery or from mainboard. Any suggestions?
You can use a DC/DC step-down converter connected to the supply; when the AC adapter gets disconnected, the CCFL inverter will be powered through the battery instead through the converter (you can use an electromagnetic relay). The real trick is to find the Power ON signal from the motherboard and to make sure that, when you'll power OFF the laptop, the CCFL inverter will also be shut down (a transistor connected in series with the electromagnetic relay will do).
Irrespective of the solution, you must factor in the additional power consumption, so you may need a bigger AC adapter (if you have a 65W adapter, you'll need to change it for an 85W one).
I all would be much easier if i find a CCFL Macbook
What's the point of using a Macbook Air motherboard with a 20W aditional power consumption from the LCD workflow...
Or maybe it's possible to attach a Macbook Pro 17 LED display to my G4?
Well, the point was to use the original display. Fitting an original Macbook display in your G4 case will give you the best compatibility, but the cost will ramp up (you'll still need to find ways to fit the motherboard/battery in the old case and extend the cables), so you'll end up paying more money than you'll pay for a fully fledged MacBook Pro/Air, with its original case/looks.
EDIT: the consumption will not be 20W higher, more like 8-10W, but the smaller adapter will be overloaded.
Summing up your list, you'll pay 390E for the components only, components you will not be able to test upon arrival. You will need to find a way to extend the display/keyboard/pad cables, to fit the new battery in the old G4 battery enclosure, to adapt the motherboard to the old G4 case (including a way to connect the battery to the new motherboard)...
The costs are not excessive, but you can find a fully functional Macbook Air (13" variant) at 450-500E, and a Macbook Pro at 350-400E (15") / 400-450E (17"), or even cheaper if you like to bid. The modifications of an old G4 will not worth the time/investment and will be a hard sell later. It can be a great DIY project though
What is the price of a functional Air/Pro in your country?
EDIT: A1181 is a motherboard from an early MacBook (2006+), so it will not be compatible with your old G4 display.