Being prepared matters far more when you live abroad, because the systems you grew up trusting are suddenly unfamiliar. At minimum, learn the local emergency number the week you arrive, know where the nearest hospital with an emergency department is, and figure out how ambulance services actually work in your country, since in some places you call a public service and in others you go through your insurer or a private hospital.
Beyond that, keep a small folder, physical and digital, with copies of passports, visas, insurance cards, prescriptions, and emergency contacts. Register with your embassy if they offer it, as that is how they reach citizens during disasters.
Finally, talk it through with family. A five minute conversation about where to meet and whom to call removes most of the panic when something actually happens.