try tapping the screwdriver in and turn at the same time but theres only one screw holds the euro in the lock

The cylinder itself has one screw that goes through the jamb of the door into the cylinder this has to be screwed out ,the key has to be inserted into the cylinder and turned until the peg that operates the gearbox lock is lined up within the cylinder then the cylinder can be extracted
The handles do have grub screws in the side to remove the handles from the door you will only have to remove one handle so only one grub screw needs to be removed ,you can try using a flat head screw by jamming it into the grub screw if this doesn't work you will have to drill the grub screw out

I've had to deal with this issue a number times and what I've done is get a small sharp edge like a chisel and hammer then gently tap the screw head anti-clockwise which will loosen up the screw until it is loose enough to remove.

Rotherham and Doncaster's answer is the first thing we would normally try, and always drop a little WD40 on it first.

Use a dremel and cut a new deeper slot for a better purchase on the screw

There are no grub screws it’s normally a holding screw that goes through the centre of the cylinder that keeps it in place unless you have a cylinder guard on the cylinder itself if the lock itself is an old Yale lock then the holding screw is hidden many thanks Daryl

A yale 3 star has two spot welds ontop of the lock to stop the lock from being pulled. You need to take lock out from ext side after you have removed handles.

1. With a screwdriver tap it as you turn.
2 . Small chisel tap until you create a perfect gap for the flat screwdriver to grab. Also tapping.
3. Remove the handels both side and spray wd40 . Tap it until it creates a small move. Unscrew.
4. If nothing works unscrew with the unscrew kit from ebay, requires attention and precision as the hole will be perfect for drilling.
5. If not even like that it doesn work means who ever fix the cylinder was an amator and you should call me asap. Thanks