First things first. When you press the buttons does the little red light come on? If the fob isn't working then we're not going to get very far.
If the lights come on then that's a start. It just means that the alarm ECU and the fob aren't talking to each other.
That looks like the TVSS which was fitted to the 80 outside of the factory, but effectively commissioned by Toyota
It was professionally spliced into the wiring loom and a sounder fitted, typically in the wing. There is a battery in there that over time dies. It is actually replaceable with a bit of fettling but if the siren etc had had it, then frankly it's a waste of time trying.
This is one of those one step at a time fixes to find out what's going on. It could just be a fuse. People who have alarm issues get fed up and pull the fuse out to kill the system. A dying battery in the siren can cause it to go off randomly and really annoy the neighbours.
So, light on the fob yes no
If yes, try pressing and holding both button at one for a few seconds. This can sync the fob to the car.
If no, then we need to go deeper into the system
If you put the key in the driver door and turn it, it should fire all the door locks. if it doesn't then there is a central locking problem. Again maybe a fuse.
Start with that and let us know what happens