You can do a GET request to <routerurl>/rest/container first, that will get you the ".id" for your container. If the container never get deleted then .id never changes, so may be able to skip this step. The ".id" should have a star (*) and a number, like *33. This .id also should up in CLI via "/container/print show-ids"
With the .id of the container, you can then do a POST to <routerurl>/rest/container/start, with a JSON body like { "number": "*33" } — where "number" means the ".id" of the container.
FWIW , I created a schema for the REST API that's usable in stuff like Postman that helps figure some of these things out: viewtopic.php?t=199476