OH 1 doesn’t have a REST API. I think there is a URL you can use assuming you can ClassicUI installed (is that what it was called?) to command an Item but I don’t think there was a way to update Items.
You can use MQTT but I’m not sure the best way to set that up. Probably to configure the MQTT Event Bus on the OH 1.x instance and then use Marketplace MQTT Event Bus on the OH 3 side, but you’ll have to modify it to use the topic structure that the MQTT 1.x binding produced.
NOTE: I’m assuming you can even install the MQTT binding on OH 1 any more. About a year ago OH lost all the artifacts from before OH 2.5.12 so unless you happen to have a local copy of it you might be out of luck.
It might be roughly the same amount of work to update the OH 1 instance to OH 2.5.12 and then use the Remote openHAB binding.