I just bought my first bike for 80 Euros

Alright bud, congrats, this is how I got into motorcycling as well, albeit on a different make/model.

Some things to check, if you're already aware of this stuff my bad, but maybe you'll find some useful info here:

1. The tank is probably fine, I used electrolysis+washing soda to derust that fucker over an afternoon. POR15 is great to permanently block water from being a problem again after the initial derust.
2. Carbs are going to need careful cleaning, I never tried berrymans (gallon of noxious chemicals sitting around is not my vibe), but I do recommend ultrasonic cleaning. The rubber is likely degraded (pinholes, dry rot), see if you can find people making aftermarket replacements which use viton, resistant to ethanol in modern gasoline
3. For the future, never spin a motor without putting some oil into the spark plugs holes, the piston rings will gouge the fuck outta the piston walls. Gouges = reduced compression, which = lower engine power, besides burning oil. Clean around the spark plugs to prevent any grit from falling in. Pull out the spark plugs, borrow an endoscope camera from AutoZone or whatever, check inside the cylinders for rust or for circumferential gouges left by the rusting piston rings as well. Nobody ever drops a little oil into the pistons before putting them away, the small amount of humidity that sneaks in causes flash rust. You'll be honing the cylinder wall, it's a bit of work but not too bad.
4. Whether you hone or not, best way to check engine condition is using a compression tester, fill the engine with the right kind of oil (should meet at least JASO MA2, no friction modifiers!! They make your clutch permanently slip), plug a charged spare battery in (even a jumper from your car), hook a compression tester to one cylinder while the other cyl has its spark plug thoroughly loosened but not removed (otherwise it may jet oil out into your shop, you can also use some paper towels). Crank, crank, crank, check the tester for the max value. I don't know this bike, but look for at least 110psi compression, with approx. symmetric values on both cyls.
5. Save money and use cheap oil during restoration, the filter will catch the important debris as you keep reusing the oil, because you'll find that as you get closer and closer to proper operating status, you'll be emptying/refilling the oil to get at internals for inspection/adjustment, thus no need to keep using brand new oil. Definitely use the right stuff on the final fill though, and please recycle the old stuff and the filters at the place you bought the oil.

Plenty of other work to do, as your life is in your hands, but you'll learn a lot, and have fun too. Keep an eye out for corroded electrical connections too, there's a wiring diagram in the manual that's pretty easy to follow.

The official service manual is your friend, take the time to read the job first before going for it. Crazy I gotta say that, but I've seen other people really fuck some moto jobs up.

Have fun, keep us posted on your progress!

Just a random passerby with no motorcycle experience -

That green is awesome! Bummer that the tank will need to be replaced.

Quote “this will probably be my main project”. Yep. Everything that Rubber says plus … tank may be ok. Start by washing it out with petrol and ball bearings (take tap and filter off). Also consider adding an external filter in your new fuel lines before the carb. Gearbox may be the reason it was left to die, or may be straightforward - you will have to get it open. Resist the temptation to turn the engine over again. Post lots of photos. Have fun.

If it's you first bike, I recommend changing the front breaks to discs.

Keep original parts for when you handle traffic sufficiently well to add the difficulty of very crappy breaking.

On a similar note, at least reshine the mirrors in the front/rear lamps so people see you.

Edit: forgot the most important part: great purchase and great project.

PS: change all the hoses and gasket to models that are compatible with mixed fuel/ethanol or you're in for a very bad surprise if not trashed engine.

if you spend another 4000 it could be worth as much as 1200.

Hard/impossible to find parts. You can try and de-rust the tank, but they usually rust up quickly or if you coat them, the coating eventually flakes off.

My advice: part it out.

I'm pretty sure I know your garage x)

My own take (having plenty of old bikes in various states of disrepair) is either buy another for spares, or ditch orignality and engine swap it (ax100 is cheap and half a chance of fitting)