Israel is far behind Europe with most waste is put into landfills, low recycling rates and few facilities for producing electricity from waste.
The deal has raised speculation that Israel's Leviathan field, close to Cyprus's Aphrodite, will link up to Egypt via Cyprus.
Energean owns the rights to the Tanin and Karish gas fields as well as five new blocks recently awarded to it by the Israeli government.
Mitsubishi Hitachi Power Systems' US-based unit PW Power Systems will construct the 400-megawatt power station.
ACU Strategic Partners wants to build nuclear reactors in Arab countries and make Israel responsible for guarding them.