When I bought the car it had the key to set the alarm on and off under the hood but the remote wasn't there. I think it's a Cobra alarm and the manual had info on it too, so it must be a factory mount. I haven't tested it since I'm missing the remote, but does anyone know if the alarm gets armed just by turning the key to lock the central locking, or does it require the remote to function?

A Cobra probably requires the remote to arm it.

I presume there's some sort of "secret handshake" to get the alarm to accept a new remote, dunno if any of these would work or not eBay.co.uk - cobra fob, Car Accessories, Automobilia, Cars, Parts Vehicles items at low prices