Welcome to the board, swiss traveller! First of all, the required information I come up with are from searching Athens´airport web-page and not from personal experience.

Does LX has its own lounge there or share it with other companies ?
Swissport lounge is located in Satellite Terminal on Departure level near gate A32. I do not know how many airlines are share this location.

Can I have access to it when arriving or is it only accessible when departing ?
Only accessible when departing. To enter most lounges, you need a same-day-boarding pass.

One advice: If you ask for an upgrade and invest 5.000 miles just to have access in the lounge, look on which carrier you are flying onwards. Maybe this is on a different Terminal and this carrier uses another lounge.