Install your eSIM while you still have wifi, before you leave home or while you wait at the gate. Installing it early does not start the clock. Your plan only begins counting days once it connects to a network in Cyprus, so you can set it up days ahead with no waste.
After you buy, MegaEsim emails you a QR code. On an iPhone, open Settings, then Cellular, then Add eSIM, and scan the code. On Android, open Settings, then Network, then SIM manager, then Add eSIM. The whole thing takes about two minutes.
One quick check first. Dial *#06# on your phone. If you see an EID number, your phone supports eSIM. Most iPhones from the XS up, Samsung Galaxy S20 and newer, and Pixel 3 and newer will work fine.